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Planetary Orbit

Planetary orbit

.]] :For other meanings of the term "orbit", see orbit (disambiguation) In physics, an orbit is the path that an object makes around another object while under the influence of a source of centripetal force, such as gravity.

History

Orbits were first analysed mathematically by Johannes Kepler who formulated his results in his laws of planetary motion. He found that the orbits of the planets in our solar system are elliptical, not circular (or epicyclic), as had previously been believed. Isaac Newton demonstrated that Kepler's laws were derivable from his theory of gravitation and that, in general, the orbits of bodies responding to the force of gravity were conic sections. Newton showed that a pair of bodies follow orbits of dimensions that are in inverse proportion to their masses 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.

Planetary orbits

Within a planetary system, planets, asteroids, comets and space debris orbit the central star in elliptical orbits. Any comet in a parabolic or hyperbolic orbit about the central star is not gravitationally bound to the star and therefore is not considered part of the star's planetary system. To date, no comet has been observed in our solar system with a distinctly hyperbolic orbit. Bodies which are gravitationally bound to one of the planets in a planetary system, either natural or artificial satellites, follow orbits about that planet. Due to mutual gravitational perturbations, the eccentricities of the orbits of the planets in our solar system vary over time. Pluto and Mercury have the most eccentric orbits. At the present epoch, Mars has the next largest eccentricity while the smallest eccentricities are those of the orbits of Venus and Neptune. As an object orbits another, 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. In the elliptical orbit, the centre of mass of the orbiting-orbited system will sit at one focus of both orbits, with nothing present at the other focus. As a planet approaches periapsis, the planet will increase in velocity. As a planet approaches apoapsis, the planet will decrease in velocity. See also: Kepler's laws of planetary motion

Understanding orbits

There are a few common ways of understanding orbits.
- As the object moves sideways, it falls toward the orbited object. However it moves so quickly that the curvature of the orbited object will fall 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 falls, it moves sideways fast enough (has enough tangential velocity) to miss the orbited object. 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. As an illustration of the orbit around a planet (eg Earth), the much-used cannon model may prove useful (see image below). Imagine a cannon sitting on top of a (very) tall mountain, which fires a cannonball horizontally. The mountain needs to be very tall, so that the cannon will be above the Earth's atmosphere and we can ignore the effects of air friction on the cannon ball. 300px If the cannon fires its ball with a low initial velocity, the trajectory of the ball will curve downwards and hit the ground (A). As the firing velocity is increased, the cannonball will hit the ground further (B) and further (C) away from the cannon, because while the ball is still falling towards the ground, the ground is curving away from it (see first point, above). If the cannonball is fired with sufficient velocity, the ground will curve away from the ball at the same rate as the ball falls - it is now in orbit (D). The orbit may be circular like (D) or if the firing velocity is increased even more, the orbit may become more (E) and more (F) elliptical. At a certain even faster velocity (called the escape velocity) the motion changes from an elliptical orbit to a parabola.

Newton's laws of motion

For a system of only two bodies that are only influenced by their mutual gravity, their orbits can be exactly calculated by Newton's laws of motion and gravity. Briefly, the sum of the forces will equal the mass times its acceleration. Gravity is proportional to mass, and falls off proportionally to the square of distance. To calculate, it is convenient to describe the motion in a coordinate system that is centered on the heavier body, and we can say that the lighter body is in orbit around the heavier body. An unmoving body that's far from a large object has more energy than one that's close. This is because it can fall farther. This is called "potential energy" because it is not yet actual. With two bodies, an orbit is a flat curve. The orbit can be open (so the object never returns) or closed (returning), depending on the total 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. The path of a free-falling (orbiting) body is always a conic section. An open orbit has the shape of a hyperbola (or in the limiting case, a parabola); 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 is often the case with comets that occasionally approach the Sun. A closed orbit has the shape of an ellipse (or in the limiting case, a circle). 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 path 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. Therefore 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, synonymously, periselene and aposelene). 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 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 each planet, the ratio of the 3rd power of its semi-major axis to the 2nd power of its period is the same constant value for all planets. Except for special cases like Lagrangian points, no method is known to solve the equations of motion for a system with four or more bodies. The 2-body solutions were published by Newton in Principia in 1687. In 1912, K. F. Sundman developed a converging infinite series that solves the 3-body problem, however it converges too slowly to be of much use. Instead, orbits 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 moon, planets and other bodies are known with great accuracy, and are used to generate tables for celestial navigation. 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 ones corresponds to solving an initial value problem. Numerical methods calculate the positions and velocities of the objects a tiny time in the future, then repeat this. However, tiny arithmetic errors from the limited accuracy of a computer's math accumulate, limiting 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 objects have been simulated.

Analysis of orbital motion

(see also orbit equation and Kepler's first law) To analyse 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 centre of force. In such coordinates the radial and transverse components of the acceleration are, respectively: :

\frac - r\left( \frac \right)^2

and :

\frac\frac\left( r^2\frac \right)

. Since the force is always radial, the transverse acceleration is zero, and it follows that: :

\frac = hu^2

, where h is a constant of integration and we have introduced the auxiliary variable u defined as 1/r. If magnitude of the radial force is f(r) per unit mass of the orbiting body, then the elimination of the time variable from the radial component of the equation of motion yields: :

\frac + u = \frac

. In the case of an 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 equation of the orbit described by the particle is thus: :

r = \frac = \frac

, where φ and e are constants of integration and L is the Semi-latus rectum. This can be recognised as the equation of a conic section in polar coordinates.

Orbital parameters

See: Orbital elements For a general elliptic orbit, the relations between the axis, eccentricity, and least and largest distance are: :Semimajor axis = (periapsis + apoapsis)/2 = mean of the extreme radii :Periapsis = semimajor axis × (1 - eccentricity) = least distance :Apoapsis = semimajor axis × (1 + eccentricity) = largest distance Note that there are alternative definitions for a "mean radius" or "average distance": if you average the radius over time for one orbit (mean anomaly), or over the orbital angle as observed by the primary (true anomaly), then you get a different result. See here for details.

Orbital period

See: orbital period

Orbital decay

If some part of a body's orbit enters an atmosphere, its orbit can decay because of drag. At each periapsis, the object scrapes the air, 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. Eventually, the orbit circularises and then the object spirals into the atmosphere. The bounds of an atmosphere vary wildly. During solar maxima, the Earth's atmosphere causes drag up to a hundred kilometres higher than during solar minimums. Some satellites with long conductive tethers can also decay because of electromagnetic drag from the Earth's magnetic field. Basically, the wire cuts the magnetic field, and acts as a generator. The wire moves electrons from the near vacuum on one end to the near-vacuum on the other end. The orbital energy is converted to heat in the wire. 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, and so can be used indefinitely. See statite for one such proposed use. Orbital decay can also 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. Finally, 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.

Earth orbits

See Earth orbit for more details.
- Low Earth orbit
- High Earth Orbit
- Intermediate circular orbit
- Geostationary orbit
- Geosynchronous orbit
- Geostationary transfer orbit
- Molniya orbit
- Polar orbit
- Polar Sun Synchronous Orbit (this is not a complete list).

Scaling in gravity

The gravitational constant G is defined to be:
- 6.6742 × 10⁻¹¹ N·m²/kg²
- 6.6742 × 10⁻¹¹ m³/(kg·s²)
- 6.6742 × 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 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. When all densities are multiplied by four, orbits are the same, but with orbital velocities doubled. When all densities are multiplied by four, and all sizes are halved, orbits are similar, with the same orbital velocities. These properties are illustrated in the formula :

GT^2 \sigma = 3\pi \left( \frac \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.

Role in the evolution of atomic theory

When atomic structure was first probed experimentally early in the twentieth century, an early picture of the atom portrayed it as a miniature solar system bound by the coulomb force rather than by gravity. This was inconsistent with electrodynamics and the model was progressively refined as quantum theory evolved, but there is a legacy of the picture in the term orbital for the wave function of an energetically bound electron state.

External links

- An on-line orbit plotter: http://www.bridgewater.edu/departments/physics/ISAW/PlanetOrbit.html
- [http://www.braeunig.us/space/orbmech.htm Orbital Mechanics] (Rocket and Space Technology) Category:Celestial mechanics Category:Solar System als:Umlaufbahn ja:軌道 (力学) simple:Orbit th:วงโคจร

Orbit (disambiguation)

The word orbit can mean more than one thing:
- Planetary orbit
- Orbit (anatomy) - the socket in the skull which accommodates an eye
- In computing:
- ORBit - an object request broker (ORB) for CORBA
- ORBit2 - an object request broker (ORB) for CORBA
- In mathematics:
- Orbit (dynamics)
- Orbit (group theory)
- Wrigley's Orbit, a brand of chewing gum.
- Orbit (band)
- Orbit (irrigation)
- Orbit (magazine) a science fiction magazine

Gravity

Gravity is the force of attraction between massive particles. Weight is determined by the mass of an object and its location in a gravitational field. While a great deal is known about the properties of gravity, the ultimate cause of the gravitational force remains an open question. General relativity is the most successful theory of gravitation to date. It postulates that mass and energy curve space-time, resulting in the phenomenon known as gravity. The effect of the bending of spacetime is often misunderstood as most people seem to prefer to think of a falling object as accelerating when the facts do not support that assumption. Skydivers do not feel any acceleration (other than from wind resistance). Gravity is acceleration.

F=\dot=m\dot+\dotv\,\!

means (if the mass is unvarying) that there must be a force that causes a mass to accelerate. For a rocket ship, that is the rocket engine. For the earth, it is the compression of the mass between something on the surface of the earth and the earth's center of mass. The acceleration is in relation to spacetime in that the weight one feels is one's resistance to deviating from one's path in spacetime. The same holds true in the rocket ship except that a rocket engine supplies the force to accelerate an occupant from his spacetime path. There is no difference between the weight he feels because of gravity or the rocket.

Newton's law of universal gravitation

Newton's law of universal gravitation states the following: :Every object in the Universe attracts every other object with a force directed along the line of centers of mass for the two objects. This force is proportional to the product of their masses and inversely proportional to the square of the separation between the centers of mass of the two objects. Given that the force is along the line through the two masses, the law can be stated symbolically as follows. :

F = - G \frac

where: :F is the magnitude of the (repulsive) gravitational force between two objects :G is the gravitational constant, that is approximately : G = 6.67 × 10⁻¹¹ N m² kg^-2 :m₁ is the mass of first object :m₂ is the mass of second object :r is the distance between the objects It can be seen that this repulsive force F is always negative, and this means that the net attractive force is positive. The minus sign is used to hold the same value meaning as in the Coulomb's Law, where a positive force as result means repulsion between two charges. Thus gravity is proportional to the mass of each object, but has an inverse square relationship with the distance between the centres of each mass. Strictly speaking, this law applies only to point-like objects. If the objects have spatial extent, the force has to be calculated by integrating the force (in vector form, see below) over the extents of the two bodies. It can be shown that for an object with a spherically-symmetric distribution of mass, the integral gives the same gravitational attraction on masses outside it as if the object were a point mass.¹ This law of universal gravitation was originally formulated by Isaac Newton in his work, the Principia Mathematica (1687). Professor William Whewell of Cambridge University, author of History of the Inductive Sciences (1837) stated: ::The law of gravitation is indisputably and incomparably the greatest scientific discovery ever made, whether we look at the advance which it involved, the extent of the truth disclosed, or the fundamental and satisfactory nature of this truth. [In A Treasury of Science ed. Harlow Shapley et al, Harper & Bros. NY: 1946] The history of gravitation as a physical concept is considered in more detail below.

Vector form

below Newton's law of universal gravitation can be written as a vector equation to account for the direction of the gravitational force as well as its magnitude. In this formula, quantities in bold represent vectors. :

\mathbf_ =
 G 
 \, \mathbf_

\mathbf_ =
 - G 
 \, \mathbf_

where :F₁₂ is the force on object 1 due to object 2 :G is the gravitational constant :m₁ and m₂ are the masses of the objects 1 and 2 :r₂₁ = | r₂ − r₁ | is the distance between objects 2 and 1 :

\mathbf_ \equiv \frac

is the unit vector from object 2 to 1 It can be seen, that the vector form of the equation is the same as the scalar form, except for the vector value of F and the unit vector. Also, it can be seen that F₁₂ = − F₂₁. Gravitational acceleration is given by the same formula except for one of the factors m: :

\mathbf =
  G 
  \, \mathbf

Gravitational field

The gravitational field is a vector field that describes the gravitational force an object of given mass experiences in any given place in space. It is a generalization of the vector form, which becomes particularly useful if more than 2 objects are involved (such as a rocket between the Earth and the Moon). For 2 objects (e.g. object 1 is a rocket, object 2 the Earth), we simply write

\mathbf r

instead of

\mathbf r_

and

m

instead of

m_1

and define the gravitational field

\mathbf g(\mathbf r)

as: :

\mathbf g(\mathbf r) =
 G 
 \, \mathbf

so that we can write: :

\mathbf( \mathbf r) = m \mathbf g(\mathbf r)

This formulation is independent of the objects causing the field. The field has units of force divided by mass; in SI, this is N·kg⁻¹.

Problems with Newton's theory

Although Newton's formulation of gravitation is quite accurate for most practical purposes, it has a few problems:

Theoretical concerns

- There is no prospect of identifying the mediator of gravity. Newton himself felt the inexplicable action at a distance to be unsatisfactory (see "Newton's reservations" below).
- Newton's theory requires that gravitational force is transmitted instantaneously. Given classical assumptions of the nature of space and time, this is necessary to preserve the conservation of angular momentum observed by Johannes Kepler. However, it is in direct conflict with Einstein's theory of special relativity which places an upper limit—the speed of light in vacuum—on the velocity at which signals can be transmitted.

Disagreement with observation

- Newton's theory does not fully explain the precession of the perihelion of the orbit of the planet Mercury. There is a 43 arcsecond per century discrepancy between the Newtonian prediction (resulting from the gravitational tugs of the other planets) and the observed precession.
- The predicted deflection of light by gravity is only half as much as observations of this deflection, which were made after General Relativity was developed in 1915.
- The observed fact that gravitational and inertial masses are the same for all bodies is unexplained within Newton's system. General relativity takes this as a postulate. See equivalence principle.

Newton's reservations

It's important to understand that while Newton was able to formulate his law of gravity in his monumental work, he was deeply uncomfortable with the notion of "action at a distance" which his equations implied. He never, in his words, "assigned the cause of this power". In all other cases, he used the phenomenon of motion to explain the origin of various forces acting on bodies, but in the case of gravity, he was unable to experimentally identify the motion that produces the force of gravity. Moreover, he refused to even offer a hypothesis as to the cause of this force on grounds that to do so was contrary to sound science. He lamented the fact that "philosophers have hitherto attempted the search of nature in vain" for the source of the gravitational force, as he was convinced "by many reasons" that there were "causes hitherto unknown" that were fundamental to all the "phenomena of nature". These fundamental phenomena are still under investigation and, though hypotheses abound, the definitive answer is yet to be found. While it is true that Einstein's hypotheses are successful in explaining the effects of gravitational forces more precisely than Newton's in certain cases, he too never assigned the cause of this power, in his theories. It is said that in Einstein's equations, "matter tells space how to curve, and space tells matter how to move", but this new idea, completely foreign to the world of Newton, does not enable Einstein to assign the "cause of this power" to curve space any more than the Law of Universal Gravitation enabled Newton to assign its cause. In Newton's own words: :I wish we could derive the rest of the phenomena of nature by the same kind of reasoning from mechanical principles; for I am induced by many reasons to suspect that they may all depend upon certain forces by which the particles of bodies, by some causes hitherto unknown, are either mutually impelled towards each other, and cohere in regular figures, or are repelled and recede from each other; which forces being unknown, philosophers have hitherto attempted the search of nature in vain. If science is eventually able to discover the cause of the gravitational force, Newton's wish could eventually be fulfilled as well. It should be noted that here, the word "cause" is not being used in the same sense as "cause and effect" or "the defendant caused the victim to die". Rather, when Newton uses the word "cause," he (apparently) is referring to an "explanation". In other words, a phrase like "Newtonian gravity is the cause of planetary motion" means simply that Newtonian gravity explains the motion of the planets. See Causality and Causality (physics).

Einstein's theory of gravitation

Einstein's theory of gravitation answered the problems with Newton's theory noted above. In a revolutionary move, his theory of general relativity (1915) stated that the presence of mass, energy, and momentum causes spacetime to become curved. Because of this curvature, the paths that objects in inertial motion follow can "deviate" or change direction over time. This deviation appears to us as an acceleration towards massive objects, which Newton characterized as being gravity. In general relativity however, this acceleration or free fall is actually inertial motion. So objects in a gravitational field appear to fall at the same rate due to their being in inertial motion while the observer is the one being accelerated. (This identification of free fall and inertia is known as the Equivalence principle.) The relationship between the presence of mass/energy/momentum and the curvature of spacetime is given by the Einstein field equations. The actual shapes of spacetime are described by solutions of the Einstein field equations. In particular, the Schwarzschild solution (1916) describes the gravitational field around a spherically symmetric massive object. The geodesics of the Schwarzschild solution describe the observed behavior of objects being acted on gravitationally, including the anomalous perihelion precession of Mercury and the bending of light as it passes the Sun. Arthur Eddington found observational evidence for the bending of light passing the Sun as predicted by general relativity in 1919. Subsequent observations have confirmed Eddington's results, and observations of a pulsar which is occulted by the Sun every year have permitted this confirmation to be done to a high degree of accuracy. There have also in the years since 1919 been numerous other tests of general relativity, all of which have confirmed Einstein's theory.

Units of measurement and variations in gravity

tests of general relativity. (ESA image)]] Gravitational phenomena are measured in various units, depending on the purpose. The gravitational constant is measured in newtons times metre squared per kilogram squared. Gravitational acceleration, and acceleration in general, is measured in metres per second squared or in non-SI units such as galileos, gees, or feet per second squared. The acceleration due to gravity at the Earth's surface is approximately 9.81 m/s², more precise values depending on the location. A standard value of the Earth's gravitational acceleration has been adopted, called g_n. When the typical range of interesting values is from zero to tens of metres per second squared, as in aircraft, acceleration is often stated in multiples of g_n. When used as a measurement unit, the standard acceleration is often called "gee", as g can be mistaken for g, the gram symbol. For other purposes, measurements in millimetres or micrometres per second squared (mm/s² or µm/s²) or in multiples of milligals or milligalileos (1 mGal = 1/1000 Gal), a non-SI unit still common in some fields such as geophysics. A related unit is the eotvos, which is a cgs unit of the gravitational gradient. Mountains and other geological features cause subtle variations in the Earth's gravitational field; the magnitude of the variation per unit distance is measured in inverse seconds squared or in eotvoses. Typical variations with time are 2 µm/s² (0.2 mGal) during a day, due to the tides, i.e. the gravity due to the Moon and the Sun. A larger variation in the effect of gravity occurs when we move from the equator to the poles. The effective force of gravity decreases as the distance from the equator decreases, due to the rotation of the Earth, and the resulting centrifugal force and flattening of the Earth. The centrifugal force causes an effective force 'up' which effectively counteracts gravity, while the flattening of the Earth causes the poles to be closer to the center of mass of the Earth. It is also related to the fact that the Earth's density changes from the surface of the planet to its centre. The sea-level gravitational acceleration is 9.780 m/s² at the equator and 9.832 m/s² at the poles, so an object will exert about 0.5% more force due to gravity at sea level at the poles than at sea level at the equator [http://curious.astro.cornell.edu/question.php?number=310].

Comparison with electromagnetic force

The gravitational interaction of protons is approximately a factor 10³⁶ weaker than the electromagnetic repulsion. This factor is independent of distance, because both interactions are inversely proportional to the square of the distance. Therefore on an atomic scale mutual gravity is negligible. However, the main interaction between common objects and the Earth and between celestial bodies is gravity, because at this scale matter is electrically neutral: even if in both bodies there were a surplus or deficit of only one electron for every 10¹⁸ protons and neutrons this would already be enough to cancel gravity (or in the case of a surplus in one and a deficit in the other: double the interaction). However, the main interactions between the charged particles in cosmic plasma (that makes up over 99% of the universe by volume), are electromagnetic forces. In terms of Planck units: the charge of a proton is 0.085, while the mass is only . From that point of view, the gravitational force is not small as such, but because masses are small. The relative weakness of gravity can be demonstrated with a small magnet picking up pieces of iron. The small magnet is able to overwhelm the gravitational interaction of the entire Earth. Similarly, when doing a chin-up, the electromagnetic interaction within your muscle cells is able to overcome the force induced by Earth on your entire body. Gravity is small unless at least one of the two bodies is large or one body is very dense and the other is close by, but the small gravitational interaction exerted by bodies of ordinary size can fairly easily be detected through experiments such as the Cavendish torsion bar experiment. Cavendish torsion bar experiment Further reading
- Jefimenko, Oleg D., "Causality, electromagnetic induction, and gravitation : a different approach to the theory of electromagnetic and gravitational fields". Star City [West Virginia] : Electret Scientific Co., c1992. ISBN 0917406095
- Heaviside, Oliver, "[http://www.as.wvu.edu/coll03/phys/www/Heavisid.htm A gravitational and electromagnetic analogy]". The Electrician, 1893.

Gravity and quantum mechanics

It is strongly believed that three of the four fundamental forces (the strong nuclear force, the weak nuclear force, and the electromagnetic force) are manifestations of a single, more fundamental force. Combining gravity with these forces of quantum mechanics to create a theory of quantum gravity is currently an important topic of research amongst physicists. General relativity is essentially a geometric theory of gravity. Quantum mechanics relies on interactions between particles, but general relativity requires no exchange of particles in its explanation of gravity. Scientists have theorized about the graviton (a messenger particle that transmits the force of gravity) for years, but have been frustrated in their attempts to find a consistent quantum theory for it. Many believe that string theory holds a great deal of promise to unify general relativity and quantum mechanics, but this promise has yet to be realized. It is notable that in general relativity gravitational radiation (which under the rules of quantum mechanics must be composed of gravitons) is only created in situations where the curvature of spacetime is oscillating, such as for co-orbiting objects. The amount of gravitational radiation emitted by the solar system and its planetary systems is far too small to measure. However, gravitational radiation has been indirectly observed as an energy loss over time in binary pulsar systems such as PSR1913+16). It is believed that neutron star mergers and black hole formation may create detectable amounts of gravitational radiation. Gravitational radiation observatories such as LIGO have been created to study the problem. No confirmed detections have been made of this hypothetical radiation, but as the science behind LIGO is refined and as the instruments themselves are endowed with greater sensitivity over the next decade, this may change.

Experimental tests of theories

Today General Relativity is accepted as the standard description of gravitational phenomena. (Alternative theories of gravitation exist but are more complicated than General Relativity.) General Relativity is consistent with all currently available measurements of large-scale phenomena. For weak gravitational fields and bodies moving at slow speeds at small distances, Einstein's General Relativity gives almost exactly the same predictions as Newton's law of gravitation. Crucial experiments that justified the adoption of General Relativity over Newtonian gravity were the classical tests: the gravitational redshift, the deflection of light rays by the Sun, and the precession of the orbit of Mercury. More recent experimental confirmations of General Relativity were the (indirect) deduction of gravitational waves being emitted from orbiting binary stars, the existence of neutron stars and black holes, gravitational lensing, and the convergence of measurements in observational cosmology to an approximately flat model of the observable Universe, with a matter density parameter of approximately 30% of the critical density and a cosmological constant of approximately 70% of the critical density. The equivalence principle, the postulate of general relativity that presumes that inertial mass and gravitational mass are the same, is also under test. Past, present, and future tests are discussed in the equivalence principle section. Even to this day, scientists try to challenge General Relativity with more and more precise direct experiments. The goal of these tests is to shed light on the yet unknown relationship between Gravity and Quantum Mechanics. Space probes are used to either make very sensitive measurements over large distances, or to bring the instruments into an environment that is much more controlled than it could be on Earth. For example, in 2004 a dedicated satellite for gravity experiments, called Gravity Probe B, was launched to test general relativity's predicted frame-dragging effect, among others. Also, land-based experiments like LIGO and a host of "bar detectors" are trying to detect gravitational waves directly. A space-based hunt for gravitational waves, LISA, is in its early stages. It should be sensitive to low frequency gravitational waves from many sources, perhaps including the Big Bang. Speed of gravity: Einstein's theory of relativity predicts that the speed of gravity (defined as the speed at which changes in location of a mass are propagated to other masses) should be consistent with the speed of light. In 2002, the Fomalont-Kopeikin experiment produced measurements of the speed of gravity which matched this prediction. However, this experiment has not yet been widely peer-reviewed, and is facing criticism from those who claim that Fomalont-Kopeikin did nothing more than measure the speed of light in a convoluted manner. The Pioneer anomaly is an empirical observation that the positions of the Pioneer 10 and Pioneer 11 space probes differ very slightly from what would be expected according to known effects (gravitational or otherwise). The possibility of new physics has not been ruled out, despite very thorough investigation in search of a more prosaic explanation.

Recent Alternative theories

- Brans-Dicke theory of gravity
- Rosen bi-metric theory of gravity
- In the modified Newtonian dynamics (MOND), Mordehai Milgrom proposes a modification of Newton's Second Law of motion for small accelerations.

Historical Alternative theories

- Nikola Tesla challenged Albert Einstein's theory of relativity, announcing he was working on a Dynamic theory of gravity (which began between 1892 and 1894) and argued that a "field of force" was a better concept and focused on media with electromagnetic energy that fill all of space.
- In 1967 Andrei Sakharov proposed something similar, if not essentially identical. His theory has been adopted and promoted by Messrs. Haisch, Rueda and Puthoff who, among other things, explain that gravitational and inertial mass are identical and that high speed rotation can reduce (relative) mass. Combining these notions with those of T. T. Brown, it is relatively easy to conceive how field propulsion vehicles such as "flying saucers" could be engineered given a suitable source of power.
- Georges-Louis LeSage proposed a gravity mechanism, now commonly called LeSage gravity, based on a fluid-based explanation where a light gas fills the entire universe.

Self-gravitating system

A self-gravitating system is a system of masses kept together by mutual gravity. An example is a binary star.

Special applications of gravity

A height difference can provide a useful pressure in a liquid, as in the case of an intravenous drip or a water tower, and can even supply enough power for hydroelectricity. A weight hanging from a cable over a pulley provides a constant tension in the cable, also in the part on the other side of the pulley. pulley Dubuque, Iowa]] Molten lead, when poured into the top of a shot tower, will coalesce into a rain of spherical lead shot, first separating into droplets, forming molten spheres, and finally freezing solid, undergoing many of the same effects as meteoritic tektites, which will cool into spherical, or near-spherical shapes in free-fall. A fractionation tower can be used to manufacture some materials by separating out the material components based on their specific gravity.

Comparative gravities of different planets and Earth's moon

The standard acceleration due to gravity at the Earth's surface is, by convention, equal to 9.80665 metres per second squared. (The local acceleration of gravity varies slightly over the surface of the Earth; see gee for details.) This quantity is known variously as g_n, g_e (sometimes this is the normal equatorial value on Earth, 9.78033 m/s²), g₀, gee, or simply g (which is also used for the variable local value). The following is a list of the gravitational accelerations (in multiples of g) at the Sun, the surfaces of each of the planets in the solar system, and the Earth's moon : Note: The "surface" is taken to mean the cloud tops of the gas giants (Jupiter, Saturn, Uranus and Neptune) in the above table. It is usually specified as the location where the pressure is equal to a certain value (normally 75 kPa?). For the Sun, the "surface" is taken to mean the photosphere. Within the Earth, the gravitational field peaks at the core-mantle boundary, where it has a value of 10.7 m/s². For spherical bodies surface gravity in m/s² is 2.8 × 10⁻¹⁰ times the radius in m times the average density in kg/m³. When flying from Earth to Mars, climbing against the field of the Earth at the start is 100 000 times heavier than climbing against the force of the sun for the rest of the flight.

Mathematical equations for a falling body

These equations describe the motion of a falling body under acceleration g near the surface of the Earth. mantle Here, the acceleration of gravity is a constant, g, because in the vector equation above,

_

would be a constant vector, pointing straight down. In this case, Newton's law of gravitation simplifies to the law :F = mg The following equations ignore air resistance and the rotation of the Earth, but are usually accurate enough for heights not exceeding the tallest man-made structures. They fail to describe the Coriolis effect, for example. They are extremely accurate on the surface of the Moon, where the atmosphere is almost nil. Astronaut David Scott demonstrated this with a hammer and a feather. Galileo was the first to demonstrate and then formulate these equations. He used a ramp to study rolling balls, effectively slowing down the acceleration enough so that he could measure the time as the ball rolled down a known distance down the ramp. He used a water clock to measure the time; by using an "extremely accurate balance" to measure the amount of water, he could measure the time elapsed. ² :For Earth

\ g=9.8\, \mbox/\mbox^2 \quad

For other planets, multiply

\ g

by the ratio of the gravitational accelerations shown above. Note: "Average" means average in time. Note: Distance traveled, d, and time taken, t, must be in the same system of units as acceleration g. See dimensional analysis. To convert metres per second to kilometres per hour (km/h) multiply by 3.6, and to convert feet per second to miles per hour (mph) multiply by 0.68 (or, precisely, 15/22).

Gravitational potential

For any mass distribution there is a scalar field, the gravitational potential (a scalar potential), which is the gravitational potential energy per unit mass of a point mass, as function of position. It is

- G \int dm

where the integral is taken over all mass. Minus its gradient is the gravity field itself, and minus its Laplacian is the divergence of the gravity field, which is everywhere equal to -4πG times the local density. Thus when outside masses the potential satisfies Laplace's equation (i.e., the potential is a harmonic function), and when inside masses the potential satisfies Poisson's equation with, as right-hand side, 4πG times the local density.

Acceleration relative to the rotating Earth

The acceleration measured on the rotating surface of the Earth is not quite the same as the acceleration that is measured for a free-falling body because of the centrifugal force. In other words, the apparent acceleration in the rotating frame of reference is the total gravity vector minus a small vector toward the north-south axis of the Earth, corresponding to staying stationary in that frame reference.

History of gravitational theory

The first mathematical formulation of gravity was published in 1687 by Sir Isaac Newton. His law of universal gravitation was the standard theory of gravity until work by Albert Einstein and others on general relativity. Since calculations in general relativity are complicated, and Newtonian gravity is sufficiently accurate for calculations involving weak gravitational fields (e.g., launching rockets, projectiles, pendulums, etc.), Newton's formulae are generally preferred. Although the law of universal gravitation was first clearly and rigorously formulated by Isaac Newton, the phenomenon was observed and recorded by others. Even Ptolemy had a vague conception of a force tending toward the center of the Earth which not only kept bodies upon its surface, but in some way upheld the order of the universe. Johannes Kepler inferred that the planets move in their orbits under some influence or force exerted by the Sun; but the laws of motion were not then sufficiently developed, nor were Kepler's ideas of force sufficiently clear, to make a precise statement of the nature of the force. Christiaan Huygens and Robert Hooke, contemporaries of Newton, saw that Kepler's third law implied a force which varied inversely as the square of the distance. Newton's conceptual advance was to understand that the same force that causes a thrown rock to fall back to the Earth keeps the planets in orbit around the Sun, and the Moon in orbit around the Earth. Newton was not alone in making significant contributions to the understanding of gravity. Before Newton, Galileo Galilei corrected a common misconception, started by Aristotle, that objects with different mass fall at different rates. To Aristotle, it simply made sense that objects of different mass would fall at different rates, and that was enough for him. Galileo, however, actually tried dropping objects of different mass at the same time. Aside from differences due to friction from the air, Galileo observed that all masses accelerate the same. Using Newton's equation,

F = m a

, it is plain to us why: :

F = - = m_1a_1

The above equation says that mass

m_1

will accelerate at acceleration

a_1

under the force of gravity, but divide both sides of the equation by

m_1

and: :

a_1 =

Nowhere in the above equation does the mass of the falling body appear. When dealing with objects near the surface of a planet, the change in r divided by the initial r is so small that the acceleration due to gravity appears to be perfectly constant. The acceleration due to gravity on Earth is usually called g, and its value is about 9.82 m/s². Galileo didn't have Newton's equations, though, so his insight into gravity's proportionality to mass was invaluable, and possibly even affected Newton's formulation on how gravity works. However, across a large body, variations in

r

can create a significant tidal force.

Notes

- Note 1: Proposition 75, Theorem 35: p.956 - I.Bernard Cohen and Anne Whitman, translators: Isaac Newton, The Principia: Mathematical Principles of Natural Philosophy. Preceded by A Guide to Newton's Principia, by I.Bernard Cohen. University of California Press 1999 ISBN 0-520-08816-6 ISBN 0-520-08817-4
- Note 2: See the works of Stillman Drake, for a comprehensive study of Galileo and his times, the Scientific Revolution.
- Max Born (1924), Einstein's Theory of Relativity (The 1962 Dover edition, page 348 lists a table documenting the observed and calculated values for the precession of the perihelion of Mercury, Venus, and Earth.)

References

-
-
-

External links

- [http://einstein.stanford.edu/ Gravity Probe B Experiment]
- [http://www.hkshum.net/whatisgravity/ What Is Gravity? - Aimed for Kids 8+ ]
- [http://www.intelligent-forces.com Intelligent Forces Theory] Satirical "Anti-Gravitationalism" website Category:Introductory physics Category:Celestial mechanics ko:중력 ja:重力 ms:Graviti

Kepler's laws of planetary motion

Johannes Kepler's primary contributions to astronomy/astrophysics were his three laws of planetary motion. Kepler, a nearly blind though brilliant German mathematician, derived these laws, in part, by studying the observations of the keen-sighted Danish astronomer Tycho Brahe. The article on Johannes Kepler gives a less mathematical description of the laws, as well as a treatment of their historical and intellectual context. Sir Isaac Newton's later discovery of the laws of motion and universal gravitation depended strongly on Kepler's work. Although from the modern point of view, Kepler's laws can be seen as a consequence of Newton's laws, historically, it was the other way around: Kepler provided a mathematically distilled description of the empirical observations, which Newton then interpreted.

Kepler's first law

The orbit of a planet about a star is an ellipse with the star at one focus. There is no object at the other focus of a planet's orbit. The semimajor axis, a, is half the major axis of the ellipse. In some sense it can be regarded as the average distance between the planet and its star, but it is not the time average in a strict sense, as more time is spent near apocentre than near pericentre.

Kepler's second law

A line joining a planet and its star sweeps out equal areas during equal intervals of time. This is also known as the law of equal areas. Suppose a planet takes 1 day to travel from points A to B. During this time, an imaginary line, from the Sun to the planet, will sweep out a roughly triangular area. This same amount of area will be swept every day. As a planet travels in its elliptical orbit, its distance from the Sun will vary. As an equal area is swept during any period of time and since the distance from a planet to its orbiting star varies, one can conclude that in order for the area being swept to remain constant, a planet must vary in velocity. The physical meaning of the law is that the planet moves faster when it is closer to the sun. This is because the sun's gravity accelerates the planet as it falls toward the sun, and decelerates it on the way back out.

Kepler's third law

The square of the sidereal period of an orbiting planet is directly proportional to the cube of the orbit's semimajor axis. :

P^2 \propto a^3

:P = object's sidereal period :a = orbit's semimajor axis Thus, not only does the length of the orbit increase with distance, also the orbital speed decreases, so that the increase of the sidereal period is more than proportional. See the actual figures: attributes of major planets. This law is also known as the harmonic law.

Accuracy and limitations

As discussed below, Kepler's third law needs to be modified when the orbiting body's mass is not negligible compared to the mass of the body being orbited. However, the correction is fairly small in the case of the planets orbiting the Sun. A more serious limitation of Kepler's laws is that they assume a two-body system. This is a particularly bad approximation in the case of the Earth-Sun-Moon system, and for calculations of the Moon's orbit, Kepler's laws are far less accurate than the empirical method invented by Ptolemy hundreds of years before. Newton generalized Kepler's first law with the realization that an object moving at greater than escape velocity (e.g., some comets) would have an open parabolic or hyperbolic orbit rather than a closed elliptical one. Thus, all of the conic sections are possible orbits. The second law is still valid for open orbits (since angular momentum is still conserved), but the third law is inapplicable because the motion is not periodic. Because electrical forces also obey an inverse square law, Kepler's laws also apply to bodies interacting electrically. If the interaction is repulsive, as in Rutherford scattering, then the orbit is hyperbolic, with the center of repulsion outside the hyperbola. In the planetary model of the atom, the electrons were imagined to orbit the nucleus exactly like the planets orbiting the Sun, and Niels Bohr even went so far as to attempt to assign specific atomic energy levels to specific elliptical orbits. The Keplerian approximation is successful when the quantum-mechanical wavelengths are small compared to the size of the orbit (as in Rutherford scattering, and in some unusual artificially populated electron orbitals with very high values of the principal quantum number). Kepler's laws do not consider the emission of radiation. The emission of gravitational radiation is negligible in our solar system, but important in some stellar systems containing black holes or neutron stars (any system involving large masses combined with large accelerations). In the planetary model of the atom, the emission of electromagnetic radiation should have led to the collapse of all atoms, and this was a hint that classical physics was in need of modification. Kepler's laws don't incorporate relativity, so, for example, they don't correctly predict the precession of Mercury's orbit. The successful explanation of this non-Keplerian behavior was one of the most dramatic early successes of general relativity. Kepler's laws fail even more dramatically in the case of an object orbiting a black hole; here, general relativity predicts that many orbits —those that pass through the event horizon— are one-way streets to a dead-end region of spacetime.

Connection to Newton's laws and conservation laws

Kepler did not understand why his laws were correct; it was Isaac Newton who discovered the answer to this more than fifty years later. The second law can also be seen as a statement of conservation of angular momentum, which is a logical consequence of Newton's laws in the special case of a force that acts along the line connecting two objects.

Kepler's first law

Newton proposed that "every object in the universe attracts every other object along a line of the centres of the objects, proportional to each object's mass, and inversely proportional to the square of the distance between the objects." The following assumes that acceleration is of the form :

\frac = f(r)\hat

. Recall that in polar coordinates: :

\frac = \dot r\hat + r\dot\theta\hat

\frac = (\ddot r - r\dot\theta^2)\hat + (r\ddot\theta + 2\dot r \dot\theta)\hat

In component form, we have: :

\ddot r - r\dot\theta^2 = f(r)

r\ddot\theta + 2\dot r\dot\theta = 0

Since

\dot r = dr/dt

and

\ddot\theta=/

, the latter equation is equivalent to :

\frac = -2\frac

. When integrated, this yields :

\log\dot\theta = -2\log r + \log\ell

, :

\ell = r^2\dot\theta

, for some constant

\ell

, which can be shown to be the specific angular momentum. Now we substitute. Let: :

r = \frac

\dot r = -\frac\dot u = -\frac\frac\frac= -\ell\frac

\ddot r = -\ell\frac\frac = -\ell\dot\theta\frac= -\ell^2u^2\frac

The equation of motion in the

\hat

direction becomes: :

\frac + u = - \fracf\left(\frac\right)

f(r)=-k/r^2

, as Newton's law of gravitation claims, then: :

\frac + u = \frac

where -k is our proportionality constant. This differential equation has the general solution: :

u = A\cos(\theta-\theta_0) + \frac.

Replacing u with r and letting θ₀=0: :

r = \frac

. This is indeed the equation of a conic section with the origin at one focus. Thus, Kepler's first law is a direct result of Newton's law of gravitation and Newton's second law of motion.

Kepler's second law

Assuming Newton's laws of motion, we can show that Kepler's second law is consistent. By definition, the angular momentum

\mathbf

of a point mass with mass

m

and velocity

\mathbf

is : :

\mathbf \equiv m \mathbf \times \mathbf

. where

\mathbf

is the position vector of the particle. Since

\mathbf = \frac

, we have: :

\mathbf = \mathbf \times m\frac

taking the time derivative of both sides: :

\frac = \mathbf \times \mathbf = 0

since the cross product of parallel vectors is 0. We can now say that

|\mathbf|

is constant. The area swept out by the line joining the planet and the sun is half the area of the parallelogram formed by

\mathbf

and

d\mathbf

. :

dA = \begin\frac\end |\mathbf \times d\mathbf| = \begin\frac\end \left|\mathbf \times \fracdt\right| = \fracdt

Since

|\mathbf|

is constant, the area swept out by is also constant. Q.E.D.

Kepler's third law

Newton used the third law as one of the pieces of evidence used to build the conceptual and mathematical framework supporting his law of gravity. If we take Newton's laws of motion as given, and consider a hypothetical planet that happens to be in a perfectly circular orbit of radius r, then we have

F=mv^2/r

for the sun's force on the planet. The velocity is proportional to r/T, which by Kepler's third law varies as one over the square root of r. Substituting this into the equation for the force, we find that the gravitational force is proportional to one over r squared. (Newton's actual historical chain of reasoning is not known with certainty, because in his writing he tended to erase any traces of how he had reached his conclusions.) Reversing the direction of reasoning, we can consider this as a proof of Kepler's third law based on Newton's of gravity, and taking care of the proportionality factors that were neglected in the argument above, we have: :

T^2 = \frac \cdot r^3

where:
- T = planet's sidereal period
- r = radius of the planet's circular orbit
- G = the gravitational constant
- M = mass of the sun The same arguments can be applied to any object orbiting any other object. This discussion implicitly assumed that the planet orbits around the stationary sun, although in reality both the planet and the sun revolve around their common center of mass. Newton recognized this, and modified this third law, noting that the period is also affected by the orbiting body's mass. However typically the central body is so much more massive that the orbiting body's mass may be ignored. Newton also proved that in the case of an elliptical orbit, the semimajor axis could be substituted for the radius. The most general result is: :

T^2 = \frac \cdot a^3

where:
- T = object's sidereal period
- a = object's semimajor axis
- G = 6.67 × 10⁻¹¹ N · m²/kg² = the gravitational constant
- M = mass of one object
- m = mass of the other object For objects orbiting the sun, it can be convenient to use units of years, AU, and solar masses, so that G, 4π² and the various conversion factors cancel out. Also with m<<M we can set m+M = M, so we have simply

T^2=a^3

Proving Kepler's third law

Define point A to be the periapsis, and point B as the apoapsis of the planet when orbiting the sun. Kepler's second law states that the orbiting body will sweep out equal areas in equal quantities of time. If we now look at a very small periods of time at the moments when the planet is at points A and B, then we can approximate the area swept out as a triangle with an altitude equal to the distance between the planet and the sun, and the base equal to the time times the speed of the planet. :

\frac\cdot(1-\epsilon)a\cdot V_A\,dt=\frac\cdot(1+\epsilon)a\cdot V_B\,dt

(1-\epsilon)\cdot V_A=(1+\epsilon)\cdot V_B

V_A=V_B\cdot\frac

Using the law of conservation of energy for the total energy of the planet at points A and B, :

\frac-\frac=\frac-\frac

\frac-\frac=\frac-\frac

\frac=\frac\cdot \left ( \frac-\frac \right )

\frac=\frac\cdot \left ( \frac \right )

V_B^2 \cdot \left ( \frac\right ) ^2-V_B^2=\frac\cdot \left ( \frac \right )

V_B^2 \cdot \left ( \frac\right )=\frac

V_B^2 \cdot \left ( \frac\right )=\frac

V_B^2 \cdot 4\epsilon=\frac

V_B=\sqrt.

Now that we have

V_B

, we can find the rate at which the planet is sweeping out area in the ellipse. This rate remains constant, so we can derive it from any point we want, specifically from point B. :

\frac=\frac=\frac\cdot(1+\epsilon)a\cdot V_B=\frac\cdot(1+\epsilon)a\cdot\sqrt=
\frac\cdot\sqrt

However, the total area of the ellipse is equal to

\pi a \sqrta

. (That's the same as

\pi a b

, because

b=\sqrta

). The time the planet take out to sweep out the entire area of the ellipse equals the ellipse's area, so, :

T\cdot \frac=\pi a \sqrta

T\cdot \frac\cdot\sqrt=\pi \sqrta^2

T=\frac=\frac=
\frac\sqrt

T^2=\fraca^3.

However, if the mass m is not negligible in relation to M, then the planet will orbit the sun with the exact same velocity and position as a very small body orbiting an object of mass

M+m

(see reduced_mass). To integrate that in the above formula, M must be replaced with

M+m

, to give

T^2=\fraca^3

. Q.E.D.

Solution for the motion as a function of time

The Keplerian problem assumes an orbit with semimajor axis a, semiminor axis b, and eccentricity ε. To convert the laws into predictions, Kepler began by adding the orbit's auxiliary circle (that with the major axis as a diameter) and defined these points:
- c center of auxiliary circle and ellipse
- s sun (at one focus of ellipse);

\mboxcs=a\varepsilon

- p the planet
- z perihelion
- x is the projection of the planet to the auxiliary circle; then

\mboxsxz=\frac ba\mboxspz

- y is a point on the circle such that

\mboxcyz=\mboxsxz=\frac ba\mboxspz

and three angles measured from perihelion:
- true anomaly

T=\angle zsp

, the planet as seen from the sun
- eccentric anomaly

E=\angle zcx

, x as seen from the centre
- mean anomaly

M=\angle zcy

, y as seen from the centre image:kepler's-equation-scheme.png Then :

\mboxcxz=\mboxcxs+\mboxsxz

=\mboxcxs+\mboxcyz

\frac2E=a\varepsilon\frac a2\sin E+\frac2M

giving Kepler's equation :

M=E-\varepsilon\sin E

. To connect E and T, assume

r=\mboxsp

then :

a\varepsilon+r\cos T=a\cos E

and

r\sin T=b\sin E

r=\frac=\frac

\tan T=\frac=\frac ba\frac=\frac

which is ambiguous but useable. A better form follows by some trickery with trigonometric identities: :

\tan\frac T2=\sqrt\frac\tan\frac E2

(So far only laws of geometry have been used.) Note that

\mboxspz

is the area swept since perihelion; by the second law, that is proportional to time since perihelion. But we defined

\mboxspz=\frac ab\mboxcyz=\frac ab\frac2M

and so M is also proportional to time since perihelion—this is why it was introduced. We now have a connection between time and position in the orbit. The catch is that Kepler's equation cannot be rearranged to isolate E; going in the time-to-position direction requires an iteration (such as Newton's method) or an approximate expression, such as :

E\approx M+\left(\varepsilon-\frac18\varepsilon^3\right)\sin M+\frac12\varepsilon^2\sin 2M+\frac38\varepsilon^3\sin 3M

via the Lagrange reversion theorem. For the small ε typical of the planets (except Pluto) such series are quite accurate with only a few terms; one could even develop a series computing T directly from M.[http://info.ifpan.edu.pl/firststep/aw-works/fsII/mul/mueller.html]

External links

- Crowell, Benjamin, Conservation Laws, [http://www.lightandmatter.com/area1book2.html http://www.lightandmatter.com/area1book2.html], an online book that gives a proof of the first law without the use of calculus. Category:Celestial mechanics Category:Equations Category:Eponymous laws ko:케플러 법칙 ja:ケプラーの法則

Solar system

The solar system comprises our Sun and the retinue of celestial objects gravitationally bound to it. Traditionally, this is said to consist of the Sun, nine planets and their 158 currently known moons; however, a large number of other objects, including asteroids, meteoroids, planetoids, comets, and interplanetary dust, orbit the Sun as well. Although the term "solar system" is frequently applied to other star systems and the planetary systems which may comprise them, it should strictly refer to our system specifically: the word "solar" is derived from the Sun's Latin name, Sol (and the term sometimes appears as Solar System). When talking about another stellar system (or planetary system), including the star(s) and bodies associated with them through gravity, it is usual to shorten it to "the system" (e.g. "the Alpha Centauri system" or "the 51 Pegasi system").

Structure and layout of the solar system

The Sun (astronomical symbol ☉) is a main sequence G2 star that contains 99.86% of the system's known mass. Its two largest orbiting bodies, Jupiter and Saturn, account for 91% of the remainder (The Oort Cloud might hold a substantial percentage, but as yet its existence is unconfirmed). In broad terms, the charted regions of our solar system consist of the Sun and its planetary system: the eight bodies in relatively unique orbits (commonly called planets or major planets) and two belts of smaller objects (which can be called minor planets, planetoids, meteoroids, planetesimals or, in the case of Pluto, planets). Objects in orbit round the Sun all lie within the same shallow plane, called the ecliptic, and all orbit in the same direction. Many are in turn orbited by moons, and the largest are encircled by planetary rings of dust and other particles. The major planets are, in order, Mercury (☿), Venus (♀), Earth (♁), Mars (♂), Jupiter (♃), Saturn (♄), Uranus (♅/10px), Neptune (♆), and Pluto (♇), though Pluto's status has been thrown into question by the discovery of (see below). Eight of the nine planets are named after or derived from gods and goddesses from Greco-Roman mythology; Earth, a Germanic word, is known in many Romance languages as Terra, the Roman goddess of the Earth. Distances within the solar system are measured most often in astronomical units, or AU. 1 AU is the distance between the Earth and the Sun, or 149 598 000 kilometers. Pluto is roughly 38 AU from the Sun, while Jupiter lies at roughly 5.2 AU. For very large distances within the solar system, such as regions beyond Pluto or the orbital circumferences of planets, the terameter (Tm, one milliard kilometers) is sometimes used. Despite the fact that many diagrams (like the image at the top of this article), for practicality's sake, represent the solar system as having each orbit the same distance apart, in actuality the orbits are largely arranged geometrically, that is, each is roughly double the distance from the Sun as the one before it. Venus’s distance from the Sun is roughly double that of Mercury, Earth’s distance is roughly double that of Venus, Mars’s double that of Earth and so on. This relationship is roughly expressed in the Titius-Bode law, a mathematical formula for predicting the semi-major axes of planets in AU. In its simplest form, it is written :

a= 0.4 + 0.3\times k

where k=0,1,2,4,8,16,32,64,128. By this formulation, we would expect Mercury's orbit (k=0) to be 0.4 AU, and Mars's orbit (k=4) to be at 1.6 AU. In fact their orbits are 0.38 and 1.52 AU.Ceres, the largest asteroid, lies at k=8. This law is only a rough guide, and doesn't fit all of the planets (Neptune is far closer than predicted, though Pluto lies at Neptune's predicted orbit). As of now, there is no scientific explanation for why this law "works," and many claim it is merely a coincidence. Pluto

Origin and evolution of the solar system

The current hypothesis of solar system formation is the nebular hypothesis, first proposed in 1755 by Immanuel Kant. It states the solar system was formed from a gaseous cloud called the solar nebula. It had a diameter of 100 AU and was 2-3 times the mass of the Sun. Over time, the nebula began to collapse, possiby due to disturbance by a nearby supernova. This explosion sent shock waves into space, which squeezed the nebula, pushing more and more matter inward until gravitational forces overcame its internal gas pressure and it also began to collapse. As the nebula collapsed, it decreased in size, which in turn caused it to spin faster to conserve angular momentum. And as the competing forces associated with gravity, gas pressure, magnetic fields, and rotation acted on it, the contracting nebula began to flatten into a spinning pancake shape with a bulge at the center. When the nebula further condensed, a protostar was formed in the middle. This system was heated by the friction of the rocks colliding into each other. Lighter elements such as hydrogen and helium evaporated out of the centre and migrated to the edges of the disc, thus concentrating the heavier elements to form dust and rocks in the centre. These heavier elements clumped together to form planetesimals and protoplanets. In the outer regions of this solar nebula, ice and volatile gases were able to survive, and as a result, the inner planets are rocky and the outer planets were massive enough to capture large amounts of lighter gases, such as hydrogen and helium. After 100 million years, the pressures and densities of hydrogen in the centre of the collapsed nebula became great enough for the protosun to sustain thermonuclear fusion reactions. As a result of this, hydrogen was converted to helium, and a great amount of heat was released. 4×¹H → ⁴He + neutrinos + photons During that time, the protostar turned into the Sun and the protoplanets and planetesimals were transformed into planets. All of the planets formed in a relatively short time of a few million years.

Regions of the solar system

protostar's rotating magnetic field on the plasma in the interplanetary medium (Solar Wind) [http://quake.stanford.edu/~wso/gifs/HCS.html]. (click to enlarge) ]] According to their location, the objects in the solar system are divided into three zones: Zone I or the inner solar system, including terrestrial planets and the Main belt of asteroids; Zone II, including the giant planets, their satellites and the centaurs, and Zone III, or the outer solar system, comprising the area of the Trans-Neptunian objects including the Kuiper Belt, the Oort cloud, and the vast region in between.

Interplanetary medium

The environment in which the solar system resides is called the interplanetary medium. The Sun radiates a continuous stream of charged particles, a plasma known as solar wind, which forms a very tenuous "atmosphere" (the heliosphere), permeating the interplanetary medium in all directions for at least ten billion (10) miles (16 Tm or 16 km) into space. Small quantities of dust are also present in the interplanetary medium and are responsible for the phenomenon of zodiacal light. Some of the dust is likely interstellar dust from outside the solar system. The influence of the Sun's rotating magnetic field on the interplanetary medium creates the largest structure in the Solar System, the heliospheric current sheet.

The inner planets

The four inner or terrestrial planets are characterised by their dense, rocky makeup. They formed in the hotter regions close to the Sun, where lighter and more volatile materials evaporated, leaving only those with high melting points, such as silicates, which form the planets' solid crusts and semi-liquid mantles, and iron, which forms their cores. All have impact craters and many possess tectonic surface features, such as rift valleys and volcanoes. The four inner planets are: volcanoes
- Mercury (0.39 AU from the Sun): The closest planet to the Sun is also the smallest and most atypical of the inner planets, having no atmosphere and, to date, no observed geological activity save that produced by impacts. Its relatively large iron core suggests that it was once a much larger world whose outer mantle was sheared off in early formation by the Sun’s gravity.
- Venus (0.72 AU): The first truly terrestrial planet, Venus, like the Earth, possesses a thick silicate mantle around an iron core, as well as a substantial atmosphere and evidence of one-time internal geological activity, such as volcanoes. It is much drier than Earth, and its atmosphere is 90 times as dense as Earth’s, however, and composed overwhelmingly of carbon dioxide with traces of sulfuric acid.
- Earth/Moon (1 AU): The largest of the inner planets, Earth is also the only one to demonstrate unequivocal evidence of ongoing geological activity. Its liquid hydrosphere, unique among the terrestrials, is probably the reason why Earth is also the only planet where multi-plate tectonics has been observed, since water acts as a lubricant for subduction. Its atmosphere is radically different from the other terrestrials, having been altered by the presence of life to contain 21 percent free oxygen. Its satellite, the Moon, is sometimes considered a terrestrial planet in a co-orbit with its partner, since its orbit around the Sun never actually loops back on itself when observed from above. The Moon possesses many of the features in common with other terrestrial planets, though it lacks an iron core.
- Mars (1.5 AU): Smaller than the Earth or Venus, Mars possesses a tenuous atmosphere of carbon dioxide. Its surface, peppered with vast volcanoes and rift valleys such as Valles Marineris, shows that it was once geologically active and [http://www.universetoday.com/am/publish/mars_volcanoes_active.html recent evidence] suggests it may have continued to be so until very recently. Mars possesses two tiny moons thought to be captured asteroids.

The asteroid belt

Asteroids are objects smaller than planets that mostly occupy the orbit between Mars and Jupiter, between 2.3 and 3.3 AU from the Sun, and are composed in significant part of non-volatile minerals. The main belt contains tens of thousands (possibly millions) over 1 km across, though they can be as small as dust. Despite their large numbers, the total mass of the main asteroid belt is unlikely to be more than a thousandth that of the Earth. Asteroids with a diameter of less than 50 m are called meteoroids. The largest asteroid, Ceres, has a diameter of roughly 1000 km; large enough to be spherical, which would make it a planet by some definitions of the word. The asteroids are thought to be the remnants of a small terrestrial planet that failed to coalesce due to the gravitational interference of Jupiter. They are subdivided into asteroid groups and families based on their specific orbital characteristics. Asteroid moons are asteroids that orbit larger asteroids. They are not as clearly distinguished as planetary moons, sometimes being almost as large as their partners. Trojan asteroids are located in either of Jupiter's L₄ or L₅ points, though the term is also sometimes used for asteroids in any other planetary Lagrange point as well. The inner solar system is dusted with rogue asteroids, many of which cross the orbits of the inner planets.

The outer planets

The four outer planets, or gas giants, (sometimes called Jovian planets) are so large they collectively make up 99 percent of the mass known to orbit the Sun. Their large sizes and distance from the Sun meant they could hold on to much of the hydrogen and helium too light for the smaller and hotter terrestrial planets to retain.
- Jupiter (5.2 AU), at 318 Earth masses, is 2.5 times the mass of all the other planets put together. Its composition of largely hydrogen and helium is not very different from that of the Sun. Three of its 63 satellites, Ganymede, Io and Europa, share elements in common with the terrestrial planets, such as volcanism and internal heating. Jupiter has a faint, smoky ring.
- Saturn (9.5 AU), famous for its extensive ring system, shares many qualities in common with Jupiter, including its atmospheric composition, though it is far less massive, being only 95 Earth masses. Two of its 49 moons, Titan and Enceladus, show signs of geological activity, though they are largely made of ice. Titan is the only satellite in the solar system with a substantial atmosphere.
- Uranus (19.6 AU) and Neptune (30 AU), while having many characteristics in common with the other gas giants, are nonetheless more similar to each other than they are to Jupiter or Saturn. They are both substantially smaller, being only 14 and 17 Earth masses, respectively. Their atmospheres contain a smaller percentage of hydrogen and helium, and a higher percentage of “ices”, such as water, ammonia and methane. For this reason some astronomers suggested that they belong in their own category, “Uranian planets,” or “ice giants.” Both planets possess dark, insubstantial ring systems. Neptune’s largest moon Triton is geologically active. Centaurs are icy comet-like bodies that have less-eccentric orbits so that they remain in the region between Jupiter and Neptune. The first centaur to be discovered, 2060 Chiron, has been called a comet since it has been shown to develop a tail, or coma, just as comets do when they approach the sun.

The trans-Neptunian region

The area beyond Neptune, often referred to as the outer solar system or simply the "trans-Neptunian region", is still largely unexplored.

The Kuiper belt

This region's first formation, which actually begins inside the orbit of Neptune, is the Kuiper belt, a great ring of debris, similar to the asteroid belt but composed mainly of ice and far greater in extent, which lies between 30 to 50 AU from the Sun. This region is thought to be the place of origin for short-period comets, such as Halley's comet. Though there are estimated to be over 70,000 Kuiper belt objects with a diameter greater than 100 km, the total mass of the Kuiper belt is relatively low, perhaps equalling or just exceeding the mass of the Earth. Many Kuiper belt objects have orbits that take them outside the plane of the ecliptic.
- Pluto, the solar system's smallest planet, is considered to be part of the Kuiper Belt population. Like others in the belt, it has a relatively eccentric orbit inclined 17 degrees to the ecliptic and ranging from 29.7 AU from the Sun at perihelion to 49.5 AU at aphelion. It has a large moon (the largest in the solar system relative to its own size), called Charon, and, new observations suggest, two other, much smaller moons. Like the Earth/Moon, Pluto and Charon are often considered a double planet. A member of the traditional nine planets, Pluto's tiny mass (less than 1% of Earth's) and diameter have called this status into question. Kuiper belt objects with Pluto-like orbits are called Plutinos. Other Kuiper belt objects have resonant orbits and are grouped accordingly. The remaining Kuiper belt objects, in more "classical" orbits, are classified as Cubewanos. The Kuiper Belt has a very sharply defined edge. At around 49 AU, a sharp dropoff occurs in the number of objects observed. This dropoff is known as the "Kuiper Cliff", and as yet its cause is unknown. Some speculate that something must exist beyond the belt large enough to sweep up the remaining debris, perhaps as large as Earth or Mars. This view is still controversial, however.

The scattered disc

Overlapping the Kuiper belt but extending much further outwards is the scattered disc. Scattered disc objects are believed to have been originally native to the Kuiper belt, but were ejected into erratic orbits in the outer fringes. One particular scattered disc object, originally found in 2003 but confirmed two years later by Mike Brown, has renewed the old debate about what constitutes a planet since, though its size is not yet known, it is almost certainly larger than Pluto. It currently has no name, but has been given the provisional designation , and has been nicknamed "Xena" by its discoverers, after the television character. It has many similarities with Pluto: its orbit is highly eccentric, with a perihelion of 38.2 AU (roughly Pluto's distance from the Sun) and an aphelion of 97.6 AU, and is steeply inclined to the ecliptic plane, indeed, at 44 degrees, more so than any known object in the solar system. Like Pluto, it is believed to consist largely of rock and ice, and has a [http://www.gps.caltech.edu/%7Embrown/planetlila/moon/index.html moon]. Whether it and the largest Kuiper belt objects should be considered planets or whether instead Pluto should be reclassified as a minor planet has not yet been resolved.

A new region?

Sedna, the newly discovered Pluto-like object with a gigantic, highly elliptical 10,500-year orbit that takes it from about 76 to 928 AU, has too distant a perihelion to be a scattered member of the Kuiper Belt and could be the first in an entirely new population. is also believed to be a member of this population.

Comets

Comets are composed largely of volatile ices and have highly eccentric orbits, generally having a perihelion within the orbit of the inner planets and an aphelion far beyond Pluto. Short-period comets exist with apoapses closer than this, however, and old comets that have had most of their volatiles driven out by solar warming are often categorized as asteroids. Long period comets have orbits lasting thousands of years. Some comets with hyperbolic orbits may originate outside the solar system.

And beyond

The point at which the solar system ends and interstellar space begins is not precisely defined, since its outer boundaries are delineated by two separate forces: the solar wind and the Sun's gravity. gravity The heliosphere expands outward in a great bubble to about 95 AU, or three times the orbit of Pluto. The edge of this bubble is known as the termination shock; the point at which the solar wind collides with the opposing winds of the interstellar medium. Here the wind slows, condenses and becomes more turbulent, forming a great oval structure known as the heliosheath that looks and behaves very much like a comet's tail; extending outward for a further 40 AU at its stellar-windward side, but tailing many times that distance in the opposite direction. The outer boundary of the sheath, the heliopause, is the point at which the solar wind finally terminates, and one enters the environment of interstellar space. Beyond the heliopause, at around 230 AU, lies the bow shock, a plasma "wake" left by the Sun as it travels through the Milky Way. But even at this point, we could not be said to have left the solar system, for the Sun's gravity will still hold sway even up to the Oort cloud, the great mass of icy objects, currently hypothetical, believed to be the source for all long-period comets and to surround our solar system like a shell from 50,000 to 100,000 AU beyond the Sun, or almost halfway to the next star system. The vast majority of the solar system, therefore, is completely unknown.

Age of the solar system

Scientists estimate that the solar system is 4.6 billion years old. To calculate this figure, they examine an unstable element, w