User:Patrick/wt

Talk:List_of_astronomical_topics Talk:List_of_geography_topics Talk:Gravity Talk:Mathematical_models_in_physics Talk:Intercontinental_ballistic_missile Talk:Large_numbers Talk:Orders_of_magnitude_(mass) Talk:Number Talk:Orders_of_magnitude Talk:orbit Talk:Tsiolkovsky_rocket_equation Talk:List_of_science_topics Talk:South_Holland Talk:Spacecraft_propulsion

This page aims to list articles on Wikipedia that are related to Talk:Astronomy, Talk:Astrophysics and Talk:Cosmology. This is so that those interested in the subject can monitor changes to the pages by clicking on Related changes in the sidebar.

The list is not necessarily complete or up to date - if you see an article that should be here but isn't (or one that shouldn't be here but is), please do update the page accordingly.

Three astronomy related WikiProjects are being developed, please visit WikiProject Astronomical Objects, WikiProject Constellations, and WikiProject Telescopes.

This is a list of Talk:geography topics:

Geography of countries

 * Talk:Geography of Afghanistan
 * Talk:Geography of Albania
 * Talk:Geography of Algeria
 * Talk:Geography of American Samoa
 * Talk:Geography of Andorra
 * Talk:Geography of Angola
 * Talk:Geography of Anguilla
 * Talk:Geography of Antarctica
 * Talk:Geography of Antigua and Barbuda
 * Talk:Geography of Argentina
 * Talk:Geography of Armenia
 * Talk:Geography of Aruba
 * Talk:Geography of Azerbaijan
 * Talk:Geography of the Bahamas
 * Talk:Geography of Bahrain
 * Talk:Geography of Bangladesh
 * Talk:Geography of Barbados
 * Talk:Geography of Belarus
 * Talk:Geography of Belgium
 * Talk:Geography of Belize
 * Talk:Geography of Benin
 * Talk:Geography of Bermuda
 * Talk:Geography of Bhutan
 * Talk:Geography of Bolivia
 * Talk:Geography of Bosnia and Herzegovina
 * Talk:Geography of Botswana
 * Talk:Geography of Brazil
 * Talk:Geography of Brunei
 * Talk:Geography of Bulgaria
 * Talk:Geography of Burkina Faso
 * Talk:Geography of Burundi
 * Talk:Geography of Cape Verde
 * Talk:Geography of Canada
 * Talk:Geography of Côte d'Ivoire
 * Talk:Geography of East Timor
 * Talk:Geography of El Salvador
 * Talk:Geography of Fiji
 * Talk:Geography of French Polynesia
 * Talk:Geography of Gabon
 * Talk:Geography of Germany
 * Talk:Geography of Grenada
 * Talk:Geography of Guadeloupe
 * Talk:Geography of Georgia
 * Talk:Geography of Guam
 * Talk:Geography of Guatemala
 * Talk:Geography of Guinea-Bissau
 * Talk:Geography of Haiti
 * Talk:Geography of Honduras
 * Talk:Geography of Hong Kong
 * Talk:Geography of Iceland
 * Talk:Geography of Ireland
 * Talk:Geography of Israel
 * Talk:Geography of Italy
 * Geography of the Ivory Coast
 * Talk:Geography of Jamaica
 * Talk:Geography of Jersey
 * Talk:Geography of Kazakhstan
 * Talk:Geography of Kenya
 * Talk:Geography of Kiribati
 * Talk:Geography of Kuwait
 * Talk:Geography of Kyrgyzstan
 * Talk:Geography of Laos
 * Talk:Geography of Lebanon
 * Talk:Geography of Mexico
 * Talk:Geography of Myanmar
 * Talk:Geography of Nepal
 * Talk:Geography of New Zealand
 * Talk:Geography of Pakistan
 * Talk:Geography of Samoa
 * Talk:Geography of Sierra Leone
 * Talk:Geography of Sudan
 * Talk:Geography of São Tomé and Príncipe
 * Talk:Geography of Tanzania
 * Talk:Geography of Thailand
 * Talk:Geography of Togo
 * Talk:Geography of Tunisia
 * Talk:Geography of Ukraine
 * Talk:Geography of the United Kingdom
 * Talk:City status in the United Kingdom
 * Talk:Extreme points of the United Kingdom
 * Talk:Economic geography of the United Kingdom
 * Talk:Lakes of the United Kingdom
 * Talk:List of the British Isles
 * Talk:List of places in the United Kingdom
 * Talk:Regions of England
 * Talk:Rivers of the United Kingdom
 * Talk:Mountains of the United Kingdom
 * Talk:Geography of the United States
 * Talk:Extreme Points of the United States
 * Talk:Geography of the Eastern United States
 * Talk:Geography of the Interior United States
 * Talk:Geography of the Western United States
 * Talk:Geography of Puerto Rico
 * Talk:Regions of the United States
 * Talk:Historic regions of the United States
 * Talk:List of U.S. government designations for places
 * Talk:List of islands of the United States
 * Talk:List of valleys of the United States
 * Talk:List of mountains of the United States
 * Talk:List of North American deserts
 * Talk:Public Land Survey System
 * Talk:Geography of Western Sahara

Miscellaneous

 * Talk:Alphabetical List of Hoboken streets
 * Talk:Alphabetical list of Santa Clara, California streets
 * Talk:Thirty most populous cities in the world
 * Talk:List of ghost towns
 * Talk:List of places with fewer than ten people
 * Places inhabited by one person:
 * Talk:Ervings, New Hampshire, Talk:United States
 * Talk:New Amsterdam, Indiana, Talk:United States
 * Talk:Hibberts, Maine, Talk:United States
 * Talk:Lost Springs, Wyoming, Talk:United States
 * Talk:Rochefourchat, Drôme, Talk:France
 * Talk:Regional policy


 * This article covers the physics of gravitation. See also Talk:gravity (disambiguation).

Gravitation is the tendency of Talk:masses to move toward each other.

The first mathematical formulation of the theory of gravitation was made by Talk:Sir Isaac Newton and proved astonishingly accurate. He postulated the force of "universal gravitational attraction".

Newton's theory has now been replaced by Talk:Albert Einstein's theory of Talk:General relativity but for most purposes dealing with weak gravitational fields (for example, sending rockets to the moon or around the solar system) Newton's formulae are sufficiently accurate. For this reason Newton's law is often used and will be presented first.

Newton's law of universal gravitation


Newton's Talk:law of universal gravitation states the following:


 * Every object in the Talk:Universe attracts every other object with a Talk:force directed along the line of centers for the two objects that is Talk:proportional to the product of their masses and inversely proportional to the square of the separation between the two objects.

Considering only the magnitude of the force, and momentarily putting aside its direction, the law can be stated symbolically as follows.


 * $$F = G \frac{m_1 m_2}{r^2}$$

where


 * F is the maganitude of the gravitational force between two objects
 * m1 is the mass of first object
 * m2 is the mass of second object
 * r is the distance between the objects
 * G is the Talk:gravitational constant, that is approximately : G = 6.67 &times; 10&minus;11 N m2 kg-2

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.

This law of universal gravitation was originally formulated by Talk:Isaac Newton in his work, the Principia Mathematica (Talk:1687). The history of the gravitation as a physical concept is considered in more detail below.

Vector form


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



\mathbf{F}_{12} = G {m_1 m_2 \over r_{21}^2} \, \mathbf{\hat{r}}_{21} $$

As before, m1 and m2 are the masses of the objects 1 and 2, and G is the gravitational constant.


 * F12 is the force on object 1 due to object 2
 * r21 = | r1 &minus; r2 | is the distance between objects 1 and 2
 * $$ \mathbf{\hat{r}}_{21} \equiv \frac{\mathbf{r}_1 - \mathbf{r}_2}{\vert\mathbf{r}_1 - \mathbf{r}_2\vert} $$ is the Talk:unit vector from object 2 to 1

It can be seen that the vector form of the equation is the same as the Talk:scalar form, except for the vector value of F and the unit vector. Also, it can be seen that F12 = &minus; F21.

Gravitational acceleration is given by the same formula except for one of the factors m:



\mathbf{a} = G {m \over r^2} \, \mathbf{\hat{r}} $$

Einstein's theory of gravity
Newton's formulation of gravity is quite accurate for most practical purposes. There are a few problems with it though:
 * 1) It assumes that gravitational force is transmitted instantaneously by a posited method, "action at a distance". However, even Newton felt action at a distance to be unsatisfactory.
 * 2) Newton's Talk:model of absolute space and time was eventually contradicted by Einstein's theory of Talk:special relativity in the Talk:twentieth century. Einstein's theory of special relativity was successfully built on the backbone of the experimentally supported assumption that there exists some velocity at which signals can be transmitted, the Talk:speed of light in vacuum.
 * 3) It does not explain the Talk:precession of the Talk:perihelion of the Talk:orbit of the Talk:planet Mercury. This precession is small; the unexplained portion is on the order of one angular second per century (an arc-second per century).
 * 4) It predicts that light is deflected by gravity. However, this predicted deflection is only half as much as observations of this deflection, which were made after General Relativity was developed in Talk:1915.
 * 5) The observed fact that gravitation and inertial mass are the same (or at least proportional) for all bodies is unexplained within Newton's system. See Talk:equivalence principle.

Talk:Einstein developed a new theory called Talk:general relativity which includes a theory of gravity, published in Talk:1915. The gravitational aspect of this theory says that the presence of matter "warps" Talk:spacetime. Objects in Talk:free fall in the universe take Talk:geodesics in spacetime. A geodesic is the counterpart of a straight line in Euclidean geometry.

How spacetime curvature simulate gravitational force
The curvature of spacetime considered as a whole implies a rather complex picture that is usually treated with the tools of Talk:differential geometry and that requires the use of Talk:tensor calculus. It is possible though to understand - at least approximately - the mechanism of gravitation without tensors when the total curvature of spacetime is split into two components:


 * curvature of space
 * Talk:time dilation.

Both components of curvature are responsible for gravity according to Einstein's theory.

The effect of the first component, the curvature of space, is negligible in all cases when the velocities of objects are much smaller than speed of light and when the ratios of masses divided by the distances separating them are much smaller than a specific constant, namely the ratio of speed of light squared to Newtonian gravitational constant: $$c^2/G \approx 10^{27} kg/m$$. So for the majority of cases in the universe, and certainly for almost all cases in our solar system except precession of perihelion of Mercury and deflection of light rays in the vicinity of sun, we may treat the space as flat, as ordinary Euclidean space. It leaves us only with the gravitational time dilation as a possible reason for the illusion of "gravitational force" acting at the distance. Assuming that the masses are smaller than the distances divided by the constant above, the time dilation is tiny, but it is enough to cause Newtonian gravity as we know it.

The reason for this illusion is this: any mass in the universe modifies the rate of time in its vicinity this way that time runs slower closer to the mass and the change of time rate is controlled by an equation having exactly the same form as the equation that Newton discovered as his "Law of Universal Gravitation". The difference between them is in essence not in form since the Newtonian potential is replaced by the Einsteinian time rate $$d\tau/dt$$, where $$\tau$$ is the time at a point at vicinity of the mass (the proper time of objects at this point in space, the time that is measured by the clocks in this point) and $$t$$ is the time at observer at infinity, with the right side of the equation $$1-GM/(c^2 r) \,\!$$ staying the same as in Newtonian equation (with accuracy to irrelevant constants). Because of the same form of both equations, the path of the object that takes an extremum of proper time while traveling, and by this taking a geodesic in spacetime, is the same (with accuracy to the negligible in this case curvature of space) as the Newtonian orbit of this object around the mass. So it looks as if the path of the object were bent by some "force of attraction" between the object and the mass. Since bending of the object's path is clearly visible and the time dilation extremely difficult to notice, a (fictitious) "gravitational force" has been assumed rather than a (real, presently measured with precise enough and formerly unavailable clocks) time dilation as the reason for bending paths of objects moving in vicinity of masses.

So without any force involved into keeping the traveling object in line the object follows the Newtonian orbit in space just by following a geodesic in spacetime. This is Einstein's explanation why without any "gravitational forces" all the objects follow Newtonian orbits and at the same time why the Newtonian gravitation is the approximation of the Einsteinian gravitation.

In this way the Newton's "Law of Universal Gravitation" that looked to people who tried to interpret it as an equation describing a hypothetical "force of gravitational attraction" acting at a distance (except to Newton himself who didn't believe that "action at a distance" is possible) turned out to be really an equation describing spacetime geodesics in Euclidean space. We may say that Newton discovered the geodesic motion in spacetime and Einstein, by applying Riemannian geometry to it, extended it to curved spacetime, disclosed the hidden Newtonian physics, and made its math accurate.

How energy is conserved if no forces act at a distance
It often puzzles students of Einstein's gravity that without any force acting at distance the kinetic energy of a free falling objects changes. The puzzling question is "where is this kinetic energy coming from, when the object is moving down; or going to, when the object is moving up"? The old "gravitational field" of the "attractive force" that was considered to be a repository of this "gravitational energy" in Newton's gravity isn't any good any more since now, if "attractive force" is zero, so is the "gravitational field". We need to identify another repository for this energy.

As we know the total energy of an object is $$E = mc^2 \,\!$$, where $$m$$ is the so-called "Talk:relativistic mass", and $$c$$ is the speed of light. When an object falls "down" its kinetic energy goes up. Energy has mass and so $$m$$ goes up. However $$c^2$$ drops down by the same amount since the falling object gets into space where time is running slower (recall time dilation) and so the speed of light, as observed by the same distant observer who is seeing the increasing kinetic energy, is slower as well (that's why the speed of light is not constant in a gravitational field). If both $$m$$ and $$c^2$$ change in opposite directions by the same amount, the product (the total energy of the object) stays the same for a free falling object. That's how the conservation of energy works in Einstein's gravity.

There is one important result of Einstein's gravity: to keep the change of $$c^2$$ the same as change of $$m$$ there must be a relative increase in amount of space (space curvature) equal to the relative time dilation. It might be said that nature has to curve space by the same amount as time gets dilated because of nature's inability to create energy from nothing.

Why Einstein's gravity differs from Newton's
Einsteinian gravitation is not just a small modification of Newtonian gravity. Even in the limit in which general relativity can be well approximated by Newton's equations, the Talk:gravitational potential of the Newtonian theory only knows about the time dilation portion of the Einsteinian gravitational field. The space curvature is not found in the Newtonian framework at all. In all cases when the space curvature becomes relevant - like in close enough proximity to big enough masses, like stars or in the context of large enough velocities - the curvature of space can't be neglected and the predictions of Newtonian and Einsteinian theories start to differ markedly. Every time such a difference was measured, the Einsteinian theory was much closer to the actual observations - essentially, its predictions were always exact.

In particular the Einsteinian gravitation explained why Talk:Mercury's precession differs from Newtonian prediction: since Mercury is the closest planet to the sun it moves faster than any other planet, and also it is in more curved space than all other planets. This is reflected in the behavior of Mercury and the Einsteinian calculations predict this behavior within observational error.

The other Einsteinian prediction is bending light rays in vicinity of the sun. Since the Newtonian deflection of the ray corresponds only to the time dilation, and since it happens for the reasons explained in the previous section that the relative curvature of space must be the same as the relative time dilation, the total deflection is twice as big as its Newtonian prediction. The Einsteinian prediction being twice as big as Newtonian is again within the observational error.

Yet despite such an "elegant" simplification of physics (and simpler in physics is more elegant) as Einsteinian elimination of action at a distance, only the observational differences between theories count in science since it is very easy to be mislead by "elegance of logic". As Einstein said "the elegance should concern a tailor rather than a physicist". He also said that "things should be made as simple as possible but not any simpler".

E.g. before Talk:1998 a group of prominent gravity physicists maintained that to make Talk:Einstein's field equation even simpler requires to remove Einstein's Talk:cosmological constant from it. They advertised this constant as an "Einstein's biggest blunder" (apparently a term coined by Einstein himself). Lack of this constant in Einstein's field equation predicted a decelerating Talk:expansion of space, which in turn was strongly advocated by almost all gravity physicists at that time. It was called Talk:standard model of cosmology. Proving that the expansion is decelerating due to "tremendous gravitational attraction of all masses of the universe" (in Einsteinian theory where there is no "gravitational attraction" at all) was supposed to be the first proof ever that cosmology is science after all, since finally it would be able to predict something. A team of enthusiastic young astronomers has been appointed to confirm this prediction. In Talk:1998 the results came in. It turned out that the prediction is false: the space of our universe looks as if it were expanding at accelerating rate.

Units of measurement and variations in gravity
Gravitational phenomena are measured in various units, depending on the purpose. The Talk:gravitational constant is measured in newtons times Talk:metre squared per Talk:kilogram squared. Gravitational acceleration, and acceleration in general, is measured in Talk:metre per second squared or in galileos or gees. The acceleration due to gravity at the Earth's surface is approximately 9.81 m/s2, depending on the location. A standard value of the Earth's gravitational acceleration has been adopted, called g. When the typical range of interesting values is from zero to several thousand galileos, as in aircraft, acceleration is often stated in multiples of g. When used as a measurement unit, the standard acceleration is often called "gee", as g can be mistaken for g, the Talk:gram symbol. For other purposes, measurements in multiples of milligalileo (1/1000 galileo) are typical, as in Talk:geophysics. A related unit is the eotvos, which is the unit of the gravitational Talk:gradient. Mountains and other geological features cause subtle variations in the Earth's gravitional field; the magnitude of the variation per unit distance is measured in eotvos.

Typical variations with time are 0.2 mgal during a day, due to the Talk:tides, i.e. the gravity due to the moon and the sun.

Gravity, and the acceleration of objects near the Earth
The acceleration due to the apparent "force of gravity" that "attracts" objects to the surface of the earth is not quite the same as the acceleration that is measured for a free-falling body at the surface of the earth (in a frame at rest on the surface). This is because of the rotation of the earth, which leads (except at the poles) to a centrifugal force which slightly lessens the acceleration observed. See Talk:Coriolis effect.

Comparison with electromagnetic force
The gravitational interaction of Talk: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 gravity is electrically neutral: even if in both bodies there were a surplus or deficit of only one Talk:electron for every 1018 protons and Talk: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).

In terms of Talk:Planck units: the charge of a proton is 0.085, while the mass is only 8 &times; 10-20. 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 Talk:magnet picking up pieces of Talk:iron. The small magnet is able to overwhelm the gravitational interaction of the entire earth.

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.



Gravity and quantum mechanics
It is strongly believed that three of the four Talk:fundamental forces (the Talk:strong nuclear force, the Talk:weak nuclear force, and the Talk:electromagnetic force) are manifestations of a single, more fundamental force. Combining gravity with these forces of Talk:quantum mechanics to create a theory of Talk: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 particles in its explanation of gravity. Scientists have theorized about the Talk:graviton (a particle that transmits the force gravity) for years, but have been frustrated in their attempts to find a consistent Talk:quantum theory for it. Many believe that Talk:string theory holds a great deal of promise to unify general relativity and Talk:quantum mechanics, but this promise has yet to be realized. It never can be for obvious reasons if Einstein's theory is true, due to the non-existence of "gravitational attraction" (explained in the above section "Einstein's Theory of Gravity")

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 Talk:gravitational redshift, the deflection of light rays by the Sun, and the Talk:precession of the orbit of Mercury.

General relativity also explains the equivalence of gravitational and inertial mass, which has to be assumed in Newtonian theory.

More recent experimental confirmations of General Relativity were the (indirect) deduction of gravitational waves being emitted from orbiting Talk:binary stars, the existence of Talk:neutron stars and black holes, Talk:gravitational lensing, and the convergence of measurements in observational Talk:cosmology to an approximately flat model of the observable Talk:Universe, with a matter density parameter of approximately 30% of the Talk:critical density and a Talk:cosmological constant of approximately 70% of the critical density.

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 Talk:2004 a dedicated Talk:satellite for gravity experiments, called Talk:Gravity Probe B, was launched. Also, land-based experiments like Talk:LIGO are gearing up to possibly detect gravitational waves directly.

Talk: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.

Alternate theories

 * In the Talk:modified Newtonian dynamics (MOND), Talk:Mordehai Milgrom proposes a modification of Talk:Newton's Second Law of motion for small accelerations.
 * Talk:Nikola Tesla challenged Talk:Albert Einstein's Talk:theory of relativity, announcing he was working on a Talk:Dynamic theory of gravity (which began between 1892 and 1894) and argued that a "field of force" was a better concept and focused on mediums with electromagnetic energy that fills all of space.
 * Talk:Georges-Louis LeSage proposed a gravity mechanism, referred to as "LeSage gravity", based on a fluid-based explanation where a light gas fills the entire universe.

History
Although the law of universal gravitation was first clearly and rigorously formulated by Isaac Newton, the phenomenon was more or less seen by others. Even Talk: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. Talk: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. Talk:Christiaan Huygens and Talk: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 Talk: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, Talk:Galileo Galilei corrected a common misconception, started by Talk: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:


 * $$F = -{G m_1m_2 \over r^2} = m_1a_1$$

The above equation says that mass $$m_1$$ will accelerate at Talk:acceleration $$a_1$$ under the force of gravity, but divide both sides of the equation by $$m_1$$ and:


 * $$a_1 = {G m_2 \over r^2}$$

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 Talk:Earth is usually called g, and its value is about 9.8 m/s2 (or 32 ft/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 Talk:tidal force.

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 not comfortable with it because 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 Talk:Causality and Talk:Causality (physics).

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

Special applications of gravity
A height difference can provide a useful pressure in a liquid, as in the case of an Talk:intravenous drip and a water Talk:tower.

A weight hanging from a cable over a Talk:pulley provides a constant tension in the cable, also in the part on the other side of the pulley.

Comparative gravities of different planets
The acceleration due to gravity at the Earth's surface is, by convention, equal to 9.80665 metres per second squared. (The actual value varies slightly over the surface of the Earth; see Talk:gee for details.) This quantity is known variously as gn, ge, g0, gee, or simply g. The following is a list of the gravitational accelerations (in multiples of g) at the surfaces of each of the planets in the solar system:

Note: The "surface" is taken to mean the cloud tops of the gas giants (Jupiter, Saturn, Uranus and Neptune) in the above table.

For spherical bodies surface gravity in m/s2 is 2.8 &times; 10&minus;10 times the radius in m times the average density in kg/m3.

Flight phases
The following flight phases can be distinguished:
 * boost phase - 3 to 4 minutes (for a Talk:solid rocket shorter than for a liquid-propellant rocket); altitude at the end of this phase is 150 -200 km, typical burn-out speed is 7 km/s
 * midcourse phase - ca. 25 minutes - Talk:suborbital flight in an Talk:elliptic orbit, i.e. the orbit is part of an Talk:ellipse with vertical major axis; the Talk:apogee (halfway the midcourse phase) is at an altitude of typically ca. 1200 km; the Talk:semi-major axis is between one half of the radius of the Earth and the radius; the projection of the orbit on the Earth's surface is a Talk:great circle - the missile may release a few independent warheads, a large number of Talk:decoys, and Talk:chaff
 * Talk:reentry phase (starting at an altitude of 100 km) - 2 minutes

See also Talk:Missile Defense Agency.

History
Early ICBMs formed the basis of many space launch systems. Examples include: Atlas, Delta, Talk:Redstone_rocket, Titan, R-7, and Proton. Modern ICBMs tend to be smaller than their ancestors (due to increased accuracy and smaller and lighter warheads) and use solid fuels, making them less useful as orbital launch vehicles.

Countries beginning developing ICBMs have all used liquid propellants initially, because the technology is easier.

Modern ICBMs
Modern ICBMs typically carry Talk:multiple independently targetable reentry vehicles (MIRVs), each of which carries a separate nuclear warhead, allowing a single missile to hit multiple targets. MIRV was an outgrowth of the rapidly shrinking size and weight of modern warheads and the Strategic Arms Limitation Treaties which imposed limitations on the number of launch vehicles(Talk:SALT I and Talk:SALT II). It has also proved to be an "easy answer" to proposed deployments of ABM systems – it is far less expensive to add more warheads to an existing missile system than to build an ABM system capable of shooting down the additional warheads; hence, most ABM system proposals have been judged to be impractical. The only operational ABM systems were deployed in the 1970's, the US Safeguard ABM facility was located in North Dakota and was operational from 1975-1976. The USSR deployed its Galosh ABM system was deployed around Moscow in the 1970's and remains in service.



Modern ICBMs tend to use solid fuel, which can be stored easily for long periods of time. Liquid-fueled ICBMs were generally not kept fueled all the time, and therefore fueling the rocket was necessary before a launch. ICBMs are based either in Talk:missile silos, which offer some protection from military attack (including, the designers hope, some protection from a nuclear first strike), or on Talk:submarines, rail cars or heavy trucks, which are mobile and therefore hard to find.

The low flying, guided Talk:cruise missile is an alternative to Talk:ballistic missiles.

Land-based intercontinental ballistic missiles (ICBMs) and cruise missiles
The US Air Force currently operates just over 500 Talk:ICBMs at around 15 missile complexes located primarily in the northern Rocky Mountain states and the Dakotas. These are of the Talk:Minuteman III and Peacekeeper ICBM variants. Peacekeeper missiles are being phased out by 2005. All USAF Talk:Minuteman II missiles have been destroyed in accordance to START, and their launch silos sealed or sold to the public. To comply with the Talk:START II most US multiple independently targetable reentry vehicles, or Talk:MIRV’s, have been eliminated and replaced with single warhead missiles. However, since the abandonment of the START II treaty, the U.S. is said to be considering retaining 800 warheads on 500 missiles.

Sea-based ICBMs

 * The Talk:US Navy currently has 14 Ohio-class Talk:SSBNs deployed. Each submarine is equipped with a complement of 24 Trident missiles, eight with Talk:Trident I missiles, and ten with Talk:Trident II missiles.


 * The Talk:French Navy constantly maintains at least four active units, relying on two classes of Talk:SSBNs: the older Redoutable class, which are progressively decomissioned, and the newer Triomphant class. These carry 16 M45 missiles with TN75 warheads, and are sceduled to be upgraded to M51 nuclear missile around 2010.

Current and former US ballistic missiles

 * Atlas (SM-65, CGM-16) former ICBM launched from silo, now the rocket is used for other purposes
 * Talk:Titan I (SM-68, HGM-25A)
 * Talk:Titan II (SM-68B, LGM-25C) - former ICBM launched from silo, now the rocket is used for other purposes
 * Minuteman I (SM-80, LGM-30A/B, HSM-80)
 * Minuteman II (LGM-30F)
 * Minuteman III (LGM-30G) - launched from silo - as of Talk:June 28, Talk:2004, there are 517 Minuteman III missiles in active inventory
 * Talk:LG-118A Peacekeeper / MX (LG-118A, MX) - silo-based; 29 missiles were on alert at the beginning of 2004; all are to be removed from service by 2005.
 * Midgetman - has never been operational - launched from mobile launcher
 * Polaris - former SLBM
 * Poseidon - former SLBM
 * Trident - SLBM - Trident II (D5) was first deployed in 1990 and is planned to be deployed past 2020.

Soviet/Russian
Specific types of Soviet/Russian ICBMs include:
 * Talk:SS-6 SAPWOOD / R-7 / 8K71
 * SS-7 SADDLER / Talk:R-16
 * Talk:SS-8 SASIN
 * Talk:SS-9 SCARP
 * Talk:SS-11 SEGO
 * Talk:SS-17 SPANKER
 * Talk:SS-18 SATAN
 * Talk:SS-19 STILLETO
 * Talk:SS-20 SABER
 * Talk:SS-24 SCALPEL
 * Talk:SS-25 SICKLE
 * Talk:Topol-M (SS-27)

Ballistic missile submarines
Specific types of Talk:ballistic missile Talk:submarines include:
 * Benjamin Franklin class
 * Ohio class
 * Resolution class
 * Vanguard class
 * Typhoon class
 * Redoutable class
 * Triomphant class
 * Additional Soviet/Russian ballistic missile submarines

External link

 * Estimated Strategic Nuclear Weapons Inventories (September 2004)




 * For information on how large numbers are named in English, see Talk:names of large numbers.

Large numbers are Talk:numbers that are large compared with the numbers used in everyday life. Very large numbers often occur in fields such as Talk:mathematics, Talk:cosmology and Talk:cryptography. Sometimes people refer to numbers as being "astronomically large". However, mathematically it is easy to define numbers that are much larger than occur even in astronomy.

Writing and thinking about large numbers
Large numbers are often found in science, and Talk:scientific notation was created to handle both these large numbers and also very small numbers. 1.0 × 109, for example, means one billion, a 1 followed by nine zeros: 1,000,000,000, and 1.0 × 10-9 means one billionth, or 0.0000000001. Writing 109 instead of nine zeros saves the reader the effort and hazard of counting a long string of zeros to see how large the number is.

Adding a 0 to a large number multiplies it by ten: 100 is ten times 10. In scientific notation, however, the exponent only increases by one, from 101 to 102. Remember then, when reading numbers in scientific notation, that small changes in the exponent equate to large changes in the number itself: 2.5 × 105 dollars ($250,000) is a common price for new homes in the U.S., while 2.5 × 1010 dollars ($25 billion) would make you one of the world's richest people.

Large numbers in the everyday world
Some large numbers apply to things in the everyday world.

Examples of large numbers describing everyday real-world objects are:
 * cigarettes smoked in the Talk:United States in one year, on the order of 1012 (one trillion)
 * bits on a computer hard disk (typically 1012 to 1013)
 * number of cells in the human body > 1014
 * number of neuron connections in the human brain, 1014 (estimated)
 * Talk:Avogadro's number, approximately 6.022 × 1023

Other examples are given in Talk:Orders of magnitude (numbers).

Large numbers and computers
Talk:Moore's Law, generally speaking, estimates that computers double in speed about every 18 months. This sometimes leads people to believe that eventually, computers will be able to solve any mathematical problem, no matter how complicated. This is not the case; computers are fundamentally limited by the constraints of physics, and certain upper bounds on what we can expect can be reasonably formulated.

First, a rule of thumb for converting between scientific notation and powers of two, since computer-related quantities are frequently stated in powers of two. Since the logarithm of 10 in base 2 is a little more than 3, multiplying a scientific notation exponent by 3 gives its approximate value as an exponent with a base of 2. For example, 103 (1000) is somewhere in the neighborhood of 29 (512). (But remember that when dealing with very large numbers, such "neighborhoods" will themselves be quite large).

Between 1980 and 2000, hard disk sizes increased from about 10 megabytes (1 × 107) to over 100 gigabytes (1 × 1011). A 100 gigabyte disk could store the names of all of Earth's six billion inhabitants without using data compression. But what about a dictionary-on-disk storing all possible passwords containing up to 40 characters? Assuming each character equals one byte, there are about 2320 such passwords, which is about 2 × 1096. This paper points out that if every particle in the universe could be used as part of a huge computer, it could store only about 1090 bits, less than one millionth of the size our dictionary would require.

Of course, even if computers can't store all possible 40 character strings, they can easily programmed to start creating and displaying them one at a time. As long as we don't try to store all the output, our program could run indefinitely. Assuming a modern PC could output 1 billion strings per second, it would take one billionth of 2 × 1096 seconds, or 2 × 1087 seconds to complete its task, which is about 6 × 1079 years. By contrast, the universe is estimated to be 13.7 billion (1.37 × 1010) years old. Of course, computers will presumably continue to get faster, but the same paper mentioned before estimates that the entire universe functioning as a giant computer could have performed no more than 10120 operations since the Talk:big bang. This is trillions of times more computation than is required for our string-displaying problem, but simply by raising the stakes to printing all 50 character strings instead of all 40 character strings we can outstrip the estimated computational potential of even the universe itself.

Problems like our simple string-displaying example grow exponentially in the number of computations they require, and are one reason why exponentially difficult problems are called "intractible" in computer science: for even small numbers like the 40 or 50 characters we used in our example, the number of computations required exceeds even theoretical limits on mankind's computing power. The traditional division between "easy" and "hard" problems is thus drawn between programs that do and do not require exponentially increasing resources to execute.

Such limits work to our advantage in Talk:cryptography, since we can safely assume that any Talk:cipher-breaking technique which requires more than, say, the 10120 operations mentioned before will never be feasible. Of course, many ciphers have been broken by finding efficient techniques which require only modest amounts of computing power and exploit weaknesses unknown to the cipher's designer. Likewise, much of the research throughout all branches of computer science focuses on finding new, efficient solutions to problems that work with far fewer resources than are required by a naive solution. For example, one way of finding the Talk:greatest common divisor between two 1000 digit numbers is to compute all their factors by trial division. This will take up to 2 × 10500 division operations, far too large to contemplate. But the Talk:Euclidean algorithm, using a much more efficient technique, takes only a fraction of a second to compute the GCD for even huge numbers such as these.

As a general rule, then, PCs in 2004 can perform 240 calculations in a few minutes. A few thousand PCs working for a few years could solve a problem requiring 264 calculations, but no amount of traditional computing power will solve a problem requiring 2128 operations (which is about what would be required to break the 128-bit SSL commonly used in web browsers, assuming the underlying ciphers remain secure). Limits on computer storage are comparable. Talk:Quantum computers may allow certain problems to become feasible, but as of 2004 it is far too soon to tell.

"Astronomically large" numbers
Talk:ja:&

Other large numbers are found in Talk:astronomy:
 * number of atoms in the visible universe, perhaps 1079 to 1081, see Mass, Size, and Density of the Universe
 * for astronomical numbers related to distance and time, see:
 * Talk:orders of magnitude (length)
 * Talk:orders of magnitude (time)

Large numbers are found in fields such as Talk:mathematics and Talk:cryptography.

The Talk:MD5 Talk:hash function generates 128-bit results. There are thus 2128 (approximately 3.402&times;1038) possible MD5 hash values. If the MD5 function is a good hash function, the chance of a document having a particular hash value is 2-128, a value that can be regarded as equivalent to zero for most practical purposes. (But see Talk:birthday paradox.)

However, this is still a small number compared with the estimated number of Talk:atoms in the Talk:Earth, still less compared with the estimated number of atoms in the Talk:observable universe.

Even larger numbers
Talk:Combinatorial processes rapidly generate even larger numbers. The Talk:factorial function, which defines the number of Talk:permutations of a set of unique objects, grows very rapidly with the number of objects.

Combinatorial processes generate very large numbers in Talk:statistical mechanics. These numbers are so large that they are typically only referred to using their Talk:logarithms.

Talk:Gödel numbers, and similar numbers used to represent bit-strings in Talk:algorithmic information theory are very large, even for mathematical statements of reasonable length. However, some pathological numbers are even larger than the Gödel numbers of typical mathematical propositions.

Examples:
 * Talk:googol = $$10^{100}$$
 * Talk:googolplex = $$10^{\mbox{googol}}=10^{\,\!10^{100}}=\mbox{googol}^{10^{98}}$$ It is the number of states a system can be in that consists of $$10^{98}$$ particles which can each be in googol states.


 * Talk:centillion = $$10^{303}$$ or $$10^{600}$$, depending on number naming system
 * Talk:Skewes' numbers: the first is ca. $$10^{\,\!10^{10^{34}}}$$, the second $$10^{\,\!10^{10^{1000}}}$$

The total amount of printed material in the world is 1.6 &times; 1018 bits, therefore the contents can be represented by a number which is ca. $$2^{1.6 \times 10^{18}}\approx 10^{4.8 \times 10^{17}}$$

For a "power tower", the most relevant for the value are the height and the last few values. Compare with googolplex:


 * $$10^{\,\!100^{10}}=10^{10^{20}}$$
 * $$100^{\,\!10^{10}}=10^{10^{10.3}}$$

Also compare: The first number is much larger than the second, due to the larger height of the power tower, and in spite of the small numbers 1.1 (however, if these numbers are made 1 or less, that greatly changes the result). Comparing the last number with $$10^{\,\!10^{10}}$$, in the number 3000.48, the 1000 originates from the third number 1000 in the original power tower, a factor 3 comes from the second number 1000, and the minor term 0.48 comes from the first number 1000.
 * $$1.1^{\,\!1.1^{1.1^{1000}}}=10^{10^{1.02*10^{40}}}$$
 * $$1000^{\,\!1000^{1000}}=10^{10^{3000.48}}$$

A very large number written with just three digits and ordinary exponentiation is $$9^{\,\!9^9} \approx 10^{369,693,100}$$.

Standardized system of writing very large numbers
A standardized way of writing very large numbers allows them to be easily sorted in increasing order, and one can get a good idea of how much larger a number is than another one.

Talk:Tetration with base 10 can be used for very round numbers, each representing an Talk:order of magnitude in a generalized sense.

Numbers in between can be expressed with a power tower of numbers 10, with at the top a regular number, possibly in scientific notation, e.g. $$10^{\,\!10^{10^{10^{10^{4.829*10^{183230}}}}}}$$, a number between $$10\uparrow\uparrow 7$$ and $$10\uparrow\uparrow 8$$ (if the exponent quite at the top is between 10 and $$10^{10}$$, like here, the number like the 7 here is the height).

If the height is too large to write out the whole power tower, a notation like $$(10\uparrow)^{183}(3.12*10^6)$$ can be used, where $$(10\uparrow)^{183}$$ denotes a Talk:functional power of the function $$f(n)=10^n$$ (the function also expressed by the suffix "-plex" as in Talk:googolplex, see the Googol family).

Various names are used for this representation:
 * base-10 exponentiated tower form
 * tetrated-scientific notation
 * incomplete (power) tower

The notation $$(10\uparrow)^{183}(3.12*10^6)$$ is in ASCII ((10^)^183)3.12e6; a proposed simplification is 10^^183@3.12e6; the notations 10^^1@3.12e6 and 10^^0@3.12e6 are not needed, one can just write 10^3.12e6 and 3.12e6.

Thus googolplex = 10^^2@100 = 10^^3@2 = 10^^4@0.301; which notation is chosen may be considered on a number-by-number basis, or uniformly. In the latter case comparing numbers is sometimes a little easier. For example, comparing 10^^2@23.8 with 10^6e23 requires the small computation 10^.8=6.3 to see that the first number is larger.

To standardize the range of the upper value (after the @), one can choose one of the ranges 0-1, 1-10, or 10-1e10:
 * In the case of the range 0-1, an even shorter notation is (here for googolplex) like 10^^3.301 (proposed by William Elliot). This is not only a notation, it provides at the same time a generalisation of 10^^x to real x>-2 (10^^4@0=10^^3, hence the integer before the point is one less than in the previous notation). This function may or may not be suitable depending on required smoothness and other properties; it is monotonically increasing and continuous, and satisfies 10^^(x+1) = 10^(10^^x), but it is only piecewise differentiable. The Talk:inverse function is a super-logarithm or hyper-logarithm, defined for all real numbers, also negative numbers. See also Extension of tetration to real numbers.
 * The range 10-1e10 brings the notation closer to ordinary scientific notation, and the notation reduces to it if the number is itself in that range (the part "10^^0@" can be dispensed with).

Another example:
 * $$2\uparrow\uparrow\uparrow 4 =

\begin{matrix} \underbrace{2_{}^{2^{{}^{.\,^{.\,^{.\,^2}}}}}}\\ \qquad\quad\ \ \ 65,536\mbox{ copies of }2 \end{matrix}\approx (10\uparrow)^{65,531}(6.0 \times 10^{19,728}) $$ (between $$10\uparrow\uparrow 65,533$$ and $$10\uparrow\uparrow 65,534$$)

The "order of magnitude" of a number (on a larger scale than usually meant), can be characterized by the number of times (say n) one has to take the $$log_{10}$$ to get a number between 1 and 10. Then the number is between $$10\uparrow\uparrow n$$ and $$10\uparrow\uparrow (n+1)$$

An obvious property that is yet worth mentioning is:
 * $$10^{(10\uparrow)^{n}x}=(10\uparrow)^{n}10^x$$

I.e., if a number x is too large for a representation $$(10\uparrow)^{n}x$$ we can make the power tower one higher, replacing x by $$log_{10}x$$, or find x from the lower-tower representation of the $$log_{10}$$ of the whole number. If the power tower would contain one or more numbers different from 10, the two approaches would lead to different results, corresponding to the fact that extending the power tower with a 10 at the bottom is then not the same as extending it with a 10 at the top (but, of course, similar remarks apply if the whole power tower consists of copies of the same number, different from 10).

If the height of the tower is not exactly given then giving a value at the top does not make sense, and a notation like $$10\uparrow\uparrow(7.21*10^8)$$ can be used.

If the value after the double arrow is a very large number itself, the above can recursively be applied on that value.

Examples:
 * $$10\uparrow\uparrow 10^{\,\!10^{10^{3.81*10^{17}}}}$$ (between $$10\uparrow\uparrow\uparrow 2$$ and $$10\uparrow\uparrow\uparrow 3$$)
 * $$10\uparrow\uparrow 10\uparrow\uparrow (10\uparrow)^{497}(9.73*10^{32})$$ (between $$10\uparrow\uparrow\uparrow 4$$ and $$10\uparrow\uparrow\uparrow 5$$)

Some large numbers which one may try to express in such standard forms include:


 * Talk:Graham's number
 * Steinhaus' Mega and Megiston, Moser's number

External link: Notable Properties of Specific Numbers (last page of a series which treats the numbers in ascending order, hence the largest numbers in the series)

Accuracy
Note that for a number $$10^p$$, one unit change in p changes the result by a factor 10. In a number like $$10^{\,\!6.2 \times 10^3}$$, with the 6.2 the result of proper rounding, the true value of the exponent may be 50 less or 50 more. Hence the result may be a factor $$10^{50}$$ too large or too small. This is seemingly an extremely poor accuracy, but for such a large number it may be considered fair. The idea that it is the relative error that counts (a large error in a large number may be relatively small and therefore acceptable), is taken a step further here: the number is so large that even a large relative error may be acceptable. Perhaps what counts is the relative error in the exponent.

Approximate arithmetic for very large numbers
In this context approximately equal may for example mean that two numbers are both written $$10^{\,\!10^{10^{10^{10^{4.829*10^{183230}}}}}}$$, with the true values instead of 4.829 being e.g. 4.8293 and 4.8288.

Hence:
 * The sum and the product of two very large numbers are both approximately equal to the larger one.
 * $$(10^a)^{\,\!10^b}=10^{a 10^b}=10^{10^{b+\log _{10} a}}$$
 * A very large number raised to a very large power is approximately equal to the larger of the following two values: the first value and 10 to the power the second. For example, for very large n we have $$n^n\approx 10^n$$ (see e.g. the computation of mega) and also $$2^n\approx 10^n$$. Thus $$2\uparrow\uparrow 65536 > 10\uparrow\uparrow 65533$$, see table.

Uncomputably large numbers
The Talk:busy beaver function &Sigma; is an example of a function which grows faster than any computable function. Its value for even relatively small input is huge. The values of &Sigma;(n) for n = 1, 2, 3, 4 are 1, 4, 6, 13. &Sigma;(5) is not known but is definitely &ge; 4098. &Sigma;(6) is at least 1.29&times;10865.

Infinite numbers
See main article Talk:cardinal number

Although all these numbers above are very large, they are all still Talk:finite. Some fields of mathematics define Talk:infinite and Talk:transfinite numbers.


 * Talk:aleph-null is the Talk:cardinality of the Talk:infinite set of the Talk:integers
 * Talk:aleph-one is the next highest cardinal number.
 * Talk:C is the cardinality of the reals. The proposition that C=Aleph-one is known as the Talk:continuum hypothesis.
 * Talk:Mahlo cardinals
 * Talk:Indescribable cardinals

Beyond all these, Talk:Georg Cantor's conception of the Talk:Absolute Infinite surely represents the absolute largest possible concept of "large number".

Notations
Some notations for extremely large numbers:
 * Talk:Knuth's up-arrow notation / Talk:hyper operators / Talk:Ackermann function, including Talk:tetration
 * Talk:Conway chained arrow notation
 * Talk:Steinhaus-Moser notation; apart from the method of construction of large numbers, this also involves a graphical notation with Talk:polygons; alternative notations, like a more conventional function notation, can also be used with the same functions.

These notations are essentially functions of integer variables, which increase very rapidly with those integers. Ever faster increasing functions can easily be constructed recursively by applying these functions with large integers as argument.

Note that a function with a vertical asymptote is not helpful in defining a very large number, although the function increases very rapidly: one has to define an argument very close to the asymptote, i.e. use a very small number, and constructing that is equivalent to constructing a very large number, e.g. the reciprocal.

Extensions
Superreal, hyperreal and surreal numbers extend the real numbers by adding infinitesimal and infinitely large numbers. While (most) real numbers have infinitely long expansions to the right of the decimal point, one can also try to allow for infinitely long expansions to the left in base p, where p is a prime, leading to the Talk:p-adic numbers. For dealing with infinite collections, the natural numbers have been generalized to the ordinal numbers and to the cardinal numbers. The former give the ordering of the collection, the latter its size. (For the finite case, the ordinal and cardinal numbers are equivalent; they diverge in the infinite case.)

The arithmetical operations of numbers, such as Talk:addition, Talk:subtraction, Talk:multiplication and Talk:division, are generalized in the branch of Talk:mathematics called Talk:abstract algebra; one obtains the groups, rings and fields.

Orders of magnitude of various quantities
In the following table the different quantities are lined up so that the following are in the same row:


 * length and the approximate time taken by light to cross that length
 * area of a square and the length of one side
 * volume of a cube and the area of one face
 * mass of some water and its volume at 4 degrees Celsius or 277.16 K

See also the separate tables for time, length, area, volume, mass, energy, power, temperature and dimensionless numbers.

* Each time shown is linked to that time. However, the time taken for light to cross the corresponding length is 3 times the time shown.

** These are the standard units but this table uses a variety of units, which can make it harder to read.

Units used in the table
The table uses units and prefixes that are commonly recognized:
 * Time:
 * Talk:femtosecond (fs)
 * Talk:nanosecond (ns)
 * Talk:microsecond (&mu;s)
 * Talk:millisecond (ms)
 * Talk:second (s)
 * Talk:hour (h)
 * Talk:day (d)
 * Talk:year (yr)
 * Length:
 * Talk:attometre (am)
 * Talk:femtometre (fm)
 * Talk:picometre (pm)
 * Talk:nanometre (nm)
 * Talk:micrometre (µm)
 * Talk:millimetre (mm)
 * Talk:centimetre (cm)
 * Talk:metre (m)
 * Talk:kilometre (km)
 * Talk:astronomical unit (AU)
 * Talk:light year (LY)
 * Area:
 * Talk:square metre (m2)
 * Talk:hectare (ha)
 * Talk:square kilometre (km2)
 * Mass:
 * Talk:gram (g)
 * Talk:kilogram (kg)
 * Talk:tonne (t)
 * Volume:
 * Talk:millilitre (ml)
 * Talk:litre (l)
 * Talk:cubic metre (m3)
 * Energy:
 * millielectronvolt (meV),
 * electronvolt (eV)
 * megaelectronvolt (MeV)
 * gigaelectronvolt (GeV)
 * teraelectronvolt (TeV)
 * Talk:erg
 * Talk:joule (J)
 * kilowatthour Talk:kWh
 * megawatthour Talk:MWh
 * gigawatthour Talk:GWh
 * Temperature:
 * nanokelvin (nK)
 * microkelvin (µK)
 * millikelvin (mK)
 * Talk:kelvin (K)

Extremely large numbers
For extremely Talk:large numbers, a generalized order of magnitude can be based on their double logarithm or super-logarithm. Rounding these downward to an integer gives categories between very "round numbers", rounding them to the nearest integer and applying the inverse function gives the "nearest" round number.

The first gives rise to the categories
 * ..., 1.023-1.26, 1.26-10, 10-1e10, 1e10-1e100, 1e100-1e1000, etc.

(the first two mentioned, and the extension to the left, may not be very useful, the two just demonstrate how the sequence mathematically continues to the left).

The second gives rise to the categories
 * negative numbers, 0-1, 1-10, 10-1e10, 1e10-10^1e10, 10^1e10-10^^4, 10^^4-10^^5, etc.

(see Talk:tetration).

The "midpoints" which determine which round number is nearer are in the first case:
 * 1.076, 2.071, 1453, 4.20e31, 1.69e316,...

and, depending on the interpolation method, in the second case
 * -.301, .5, 3.162, 1453, 1e1453, 10^1e1453, 10^^2@1e1453,...

(See notation of extremely large numbers.)

For extremely small numbers (in the sense of close to zero) neither method is suitable directly, but of course the generalized order of magnitude of the reciprocal can be considered.

Similar to the logarithmic scale one can have a double logarithmic and super-logarithmic scale. The intervals above all have the same length on them, with the "midpoints" actually midway. More generally, a point midway between two points corresponds to the Talk:generalised f-mean with f(x) the corresponding function log log x or slog x. In the case of log log x, this mean of two numbers (e.g. 2 and 16 giving 4) does not depend on the base of the logarithm, just like in the case of log x (Talk:geometric mean, 2 and 8 giving 4), but unlike in the case of log log log x (4 and 65536 giving 16 if the base is 2, but different otherwise).

History
Orbits were first analysed mathematically by Kepler who formulated his results in his laws of planetary motion. He found that the orbits of the Talk:planets in our Talk:solar system are elliptical, not circular (or epicyclic), as had previously been believed.

Talk:Isaac Newton demonstrated that Kepler's laws were derivable from his theory of Talk:gravitation and that, in general, the orbits of bodies responding to the force of Talk:gravity were Talk:conic sections. Newton showed that a pair of bodies follow orbits of dimensions that are in inverse proportion to their Talk:masses about their common Talk: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 Talk:planetary system, Talk:planets, Talk:asteroids, Talk:comets and Talk:space debris orbit the central Talk:star in Talk: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 Talk: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 object, the Talk:periapsis is that Talk:point at which the orbiting object is closest to the object being orbited and the Talk:apoapsis is that Talk:point at which the orbiting object is farthest from the object being orbited.

In the elliptical orbit, the orbited object will sit at one Talk:focus; with nothing present at the other focus. As a planet approaches Talk:periapsis, the planet will increase in Talk:velocity. As a planet approaches Talk:apoapsis, the planet will decrease in velocity.

See also: Talk:Kepler's laws of planetary motion

Understanding orbits
There are a few common ways of understanding orbits.


 * As the object moves, 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 Talk:gravity, pulls the object into a curve as it attempts to fly off in a straight line.
 * As the object falls, it moves sideways fast enough (has enough tangential Talk: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 three one-dimensional orbits around a gravitational center.

As an illustration of the orbit around a planet (eg Talk:Earth), the much-used cannon model may prove useful. Imagine a cannon sitting on top of a (very) tall mountain, which fires a cannonball horizontally. If the cannon fires its ball with a low initial velocity, the trajectory of the ball will curve downwards and hit the ground. As the firing velocity is increased, the cannonball will hit the ground further and further 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.

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 Talk: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 Talk:energy of the system. In the case of an open orbit, the speed at any position of the orbit is at least the Talk: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 Talk:conic section.

An open orbit has the shape of a Talk:hyperbola (or in the limiting case, a Talk: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 Talk:comets that occasionally approach the Sun.

A closed orbit has the shape of an Talk:ellipse (or in the limiting case, a Talk:circle). The point where the orbiting body is closest to Earth is the Talk:perigee, called Talk: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 Talk:apogee, Talk: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:


 * 1) The orbit of a planet around the Sun is an Talk: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 Talk:periapsis.  The point farthest from the attracting body is called the Talk: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 Talk:Moon have a perilune and apolune (or, synonymously, periselene and aposelene). An orbit around any Talk:star, not just the Sun, has a periastron and an apastron
 * 2) 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."
 * 3) For each planet, the ratio of the 3rd power of its average distance to the Sun, to the 2nd power of its period, is the same constant value for all planets.

Except for special cases like Talk: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 Talk: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 Talk: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 Talk:celestial navigation.

The Talk: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 Talk: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 Talk: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 Talk:polar coordinates with the origin coinciding with the centre of force. In such coordinates the radial and transverse components of the Talk:acceleration are, respectively:
 * $$\frac{d^2r}{dt^2} - r\left( \frac{d\theta}{dt} \right)^2$$

and
 * $$\frac{1}{r}\frac{d}{dt}\left( r^2\frac{d\theta}{dt} \right)$$.

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


 * $$\frac{d\theta}{dt} = hu^2$$,

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:


 * $$\frac{d^2u}{d\theta^2} + u = \frac{f(1 / u)}{h^2u^2}$$.

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 (upto a shift of origin of the dependent variable).

The equation of the orbit described by the particle is thus:


 * $$r = \frac{1}{u} = \frac{l}{1 + e \cos (\theta - \phi)}$$,

where &phi; and e are constants of integration. This can be recognised as the equation of a Talk:conic section in polar coordinates.

Orbital parameters
See: Talk:Orbital elements

For a general elliptic orbit, the relations between the axis, eccentricity, and least and largest distance are:


 * Semimajor axis = (periapsis + apoapsis)/2 = geometric mean radius


 * Periapsis = semimajor axis &times; (1 - eccentricity) = least distance


 * Apoapsis = semimajor axis &times; (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, or over the central angle (true anomaly), then the average distance is a function of both semimajor axis and eccentricity. See here for details.

Orbital period
See: Talk:orbital period

Orbital decay
If some part of a body's orbit enters an atmosphere, its orbit can decay because of drag. 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 Talk: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 Talk:solar sails or Talk:magnetic sails. These forms of propulsion require no propellant or energy input, and so can be used indefinitely. See Talk:statite for one such proposed use.

Orbital decay can also occur due to Talk:tidal forces for objects below the Talk:synchronous orbit for the body they're orbiting. The gravity of the orbiting object raises Talk: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 Talk: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 Talk: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 Talk:black holes or Talk:neutron stars that are orbiting each other closely.

Earth orbits
See Talk:Earth orbit for more details.

(this is not a complete list).
 * Talk:Low Earth orbit
 * Talk:High Earth Orbit
 * Talk:Intermediate circular orbit
 * Talk:Geostationary orbit
 * Talk:Geosynchronous orbit
 * Talk:Geostationary transfer orbit
 * Talk:Molniya orbit
 * Talk:Polar orbit
 * Talk:Polar Sun Synchronous Orbit

Scaling in gravity
G = 6.6742 &times; 10&minus;11 N&middot;m2/kg2 = 6.6742 &times; 10&minus;11 m3/(kg.s2) = 6.6742 &times; 10&minus;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 Talk: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 Talk:orbital periods remains 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


 * $$ GT^2 \sigma = 3\pi \left( \frac{a}{r} \right)^3, $$

for an elliptical orbit with Talk:semi-major axis a, of a small body around a spherical body with radius r and average density &sigma;, where T is the Talk: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 Talk:atom portrayed it as a miniature solar system bound by the coulomb force rather than by gravity. This was inconsistent with Talk:electrodynamics and the model was progressively refined as Talk:quantum theory evolved, but there is a legacy of the picture in the term orbital for the wave function of an energetically bound Talk:electron state.

Energy
In the ideal case $$m_1$$ is useful payload and $$m_0-m_1$$ is reaction mass (this corresponds to empty tanks having no mass, etc.). The energy required can simply be computed as


 * $$\frac{1}{2}(m_0-m_1)v_e^2$$

Seemingly this is just the kinetic energy of the reaction mass and not the kinetic energy required for the payload, but if e.g $$v_e$$=10 km/s and the speed of the rocket is 3 km/s, then the speed of the reaction mass changes only from 3 to 7 km/s; the energy thus "saved" corresponds to the increase of the specific kinetic energy (kinetic energy per kg) for the rocket. In general:

$$d(\frac{1}{2}v^2)=vdv=vv_edm/m=\frac{1}{2}(v_e^2-(v-v_e)^2+v^2)dm/m$$

Thus $$\Delta \epsilon = \int v\, d (\Delta v)$$

where $$\epsilon$$ is the specific energy of the rocket (potential plus kinetic energy) and $$\Delta v$$ is a separate variable, not just the change in $$v$$. In the case of using the rocket for deceleration, i.e. expelling reaction mass in the direction of the velocity, $$v$$ should be taken negative.

The formula is for the ideal case again, with no energy lost on heat, etc. The latter causes a reduction of thrust, so it is a disadvantage even when the objective is to lose energy (deceleration).

If the energy is produced by the mass itself, as in a chemical rocket, the Talk:fuel value has to be :$$\frac{1}{2}v_e^2$$, where for the fuel value also the mass of the oxidizer has to be taken into account. A typical value is $$v_e=4.5 km/s$$, corresponding to a fuel value of 10.1 MJ/kg. The actual fuel value is higher, but part of the energy is lost on heat that flows off as radiation.

The required energy is


 * $$E = \frac{1}{2}m_1(e^{\Delta v\ / v_e}-1)v_e^2$$

Conclusions:


 * for $$\Delta v <<v_e$$ we have $$E\approx \frac{1}{2}m_1 v_e \Delta v $$
 * for a given $$\Delta v$$, the minimum energy is needed if $$v_e=0.6275 \Delta v$$, requiring an energy of
 * $$E = 0.772 m_1(\Delta v)^2$$.
 * Starting from zero speed this is 54.4 % more than just the kinetic energy of the payload. Starting from a nonzero speed the required energy may be less than the increase in energy in the payload. This can be the case when the reaction mass has a lower speed after being expelled than before. For example, from a LEO of 300 km altitude to an escape orbit is an increase of 29.8 MJ/kg, which, using a specific impulse of 4.5 km/s, has a net cost of 20.6 MJ/kg ($$\Delta v$$ = 3.20 km/s; the energies are per kg payload).

This optimization does not take into account the masses of various kinds of engines.

Also, for a given objective such as moving from one orbit to another, the required $$\Delta v$$ may depend greatly on the rate at which the engine can produce $$\Delta v$$ and maneuvers may even be impossible if that rate is too low. For example, a launch to LEO normally requires a $$\Delta v$$ of ca. 9.5 km/s (mostly for the speed to be acquired), but if the engine could produce $$\Delta v$$ at a rate of only slightly more than g, it would be a slow launch requiring altogether a very large $$\Delta v$$ (think of hovering without making any progress in speed or altitude, it would cost a $$\Delta v$$ of 9.8 m/s each second). If the possible rate is only $$g$$ or less, the maneuver can not be carried out at all with this engine.

The power is given by
 * $$P= \frac{1}{2}m v_e a = \frac{1}{2} v_e F $$

where $$F$$ is the thrust and $$a$$ the acceleration due to it. Thus the theoretically possible thrust per unit power is 2 divided by the specific impulse in m/s. The thrust efficiency is the actual thrust as percentage of this.

If e.g. Talk:solar power is used this restricts $$a$$; in the case of a large $$v_e$$ the possible acceleration is inversely proportional to it, hence the time to reach a required delta-v is proportional to $$v_e$$; with 100% efficiency:
 * for $$\Delta v <<v_e$$ we have $$t\approx \frac{m v_e \Delta v}{2P}$$

Examples:
 * power 1000 W, mass 100 kg, $$\Delta v$$= 5 km/s, $$v_e$$= 16 km/s, takes 1.5 months.
 * power 1000 W, mass 100 kg, $$\Delta v$$= 5 km/s, $$v_e$$= 50 km/s, takes 5 months.

Thus $$v_e$$ should not be too large.

Examples
Assume a Talk:specific impulse of 4.5 km/s and a $$\Delta v$$ of 9.7 km/s (Earth to LEO).


 * Talk:SSTO rocket: $$1-e^{-9.7/4.5}$$ = 0.884, therefore 88.4 % of the initial total mass has to be propellant. The remaining 11.6 % is for the engines, the tank, and the payload. In the case of a space shuttle, it would also include the orbiter.


 * Talk:Two stage to orbit: suppose that the first stage should provide a $$\Delta v$$ of 5.0 km/s; $$1-e^{-5.0/4.5}$$ = 0.671, therefore 67.1% of the initial total mass has to be propellant. The remaining mass is 32.9 %. After deposing of the first stage, a mass remains equal to this 32.9 %, minus the mass of the tank and engines of the first stage. Assume that this is 8 % of the initial total mass, then 24.9 % remains. The second stage should provide a $$\Delta v$$ of 4.7 km/s; $$1-e^{-4.7/4.5}$$ = 0.648, therefore 64.8% of the remaining mass has to be propellant, which is 16.2 %, and 8.7 % remains for the tank and engines of the second stage, the payload, and in the case of a space shuttle, also the orbiter. Thus together 16.7 % is available for all engines, the tanks, the payload, and the possible orbiter.

Astronomy

 * Talk:List of astronomical topics
 * Talk:Asteroids
 * Talk:List of constellations
 * ...by area
 * Talk:List of meteor showers
 * Talk:List of stars
 * Talk:List of nearest stars
 * Talk:List of brightest stars
 * Talk:List of mnemonics for star classification
 * Talk:List of semiregular variable stars
 * Talk:List of telescope types
 * Moons of the solar system
 * Talk:Timeline of astronomical maps, catalogs, and surveys
 * Talk:Timeline of cosmic microwave background astronomy
 * Talk:Timeline of cosmology
 * Talk:Timeline of galaxies, clusters of galaxies, and large-scale structure
 * Talk:Timeline of planetary exploration
 * Talk:Timeline of solar astronomy
 * Talk:Timeline of solar system astronomy
 * Talk:Timeline of stellar astronomy
 * Talk:Timeline of telescopes, observatories, and observing technology
 * Talk:Timeline of the Big Bang
 * Talk:Timeline of the interstellar medium and intergalactic medium
 * Talk:Timeline of the Universe
 * Talk:Timeline of white dwarfs, neutron stars, and supernovae
 * Talk:Zodiac

Biology

 * Talk:List of biological topics
 * Talk:List of biomolecules
 * Talk:Amino acids
 * Talk:Flower album
 * Talk:Genetic code
 * Talk:List of Acer species (maple)
 * Talk:List of Arecaceae Genera (palm tree family)
 * Talk:List of common species names
 * Talk:List of genera in Caesalpinioideae
 * Talk:List of genera in Faboideae
 * Talk:List of genera in Mimosoideae
 * Talk:List of flowers
 * Talk:List of forests
 * Talk:List of fruit
 * Talk:List of herbs
 * Talk:List of Quercus species (oak)
 * Talk:List of Smilax species
 * Talk:List of vegetables
 * Talk:Timeline of evolution

Chemistry

 * Talk:List of chemistry topics
 * Talk:List of biochemistry topics
 * Talk:Periodic table of the elements
 * Talk:List of elements by name
 * Talk:List of elements by number
 * Talk:List of elements by symbol
 * Talk:Table of standard electrode potentials
 * Talk:Isotope table (divided)
 * Talk:Isotope table (complete)
 * Talk:Functional groups
 * Talk:List of compounds

Computer Science

 * Talk:List of computing topics
 * Talk:List of Internet topics
 * Talk:List of technologies
 * Character sets and encodings
 * Talk:ASCII
 * Talk:ISO 646
 * Talk:ISO 8859-1, Talk:ISO 8859-3, Talk:ISO 8859-15
 * Talk:Unicode, Talk:UTF-8
 * Talk:List of algorithms
 * Talk:List of basic computer science topics
 * Talk:List of calculators
 * Talk:List of computer bugs
 * Talk:List of computing and IT abbreviations
 * Talk:List of data structures
 * List of Talk:graphics file formats
 * Talk:List of IBM products
 * List of Talk:internet slang
 * Talk:List of Intel microprocessors
 * Talk:List of machines running CP/M
 * Talk:List of newsgroups
 * Talk:List of operating systems
 * Lists of Talk:programming languages
 * Alphabetical list
 * Categorical list
 * Chronological list
 * Generational list
 * Talk:List of Univac products
 * Talk:List of Unices
 * List of Talk:Unix commands
 * Talk:Timeline of computing
 * Talk:Timeline of microscope technology
 * Talk:Timeline of materials technology
 * Talk:List of digital library projects

Ecology

 * Talk:List of ecological topics
 * Talk:List of Conservation topics

Geography

 * Talk:List of geography topics
 * Talk:List of places
 * Talk:List of countries
 * Talk:List of geographical pairs
 * List of twin cities
 * List of twin towns
 * List of sister cities

Geology

 * Talk:List of geology topics
 * Talk:Geologic timescale
 * List of Talk:birthstones
 * Talk:List of earthquakes
 * Talk:List of minerals
 * Talk:List of oil fields
 * Talk:Timeline of geology
 * Talk:List of volcanoes

Linguistics

 * Talk:List of linguistic topics
 * Talk:List of languages
 * Talk:Language families and languages
 * Talk:ISO 639
 * Talk:List of official languages
 * Definitions by language
 * Talk:Alphabets (and list of Talk:alphabets) and Orthography
 * Talk:Arabic alphabet
 * Talk:Aramaic alphabet
 * Talk:Braille
 * Talk:Coptic alphabet
 * Talk:Cyrillic alphabet
 * Talk:Gothic alphabet
 * Talk:Hangul (Korean alphabet)
 * Talk:Hebrew alphabet
 * Talk:Kannada language (and alphabet)
 * Talk:Hiragana, Talk:Katakana (Japanese alphabet)
 * Talk:Morse code
 * Talk:NATO phonetic alphabet
 * Talk:Phoenician alphabet
 * Talk:Runic alphabet
 * Talk:SAMPA Chart
 * Talk:SAMPA chart for English
 * Talk:Shavian alphabet
 * Talk:Thai alphabet
 * List of common misspellings
 * Talk:Pronunciation
 * Talk:List of English words without rhymes
 * Talk:List of words commonly mispronounced
 * Talk:List of words of disputed pronunciation
 * Talk:List of tongue-twisters
 * Talk:Morphemes/Word endings
 * List of Isms
 * List of Talk:Ographies
 * Talk:List of ologies
 * Talk:Acronyms and Talk:Abbreviations
 * Talk:List of acronyms
 * Wiktionary:Definitions of acronyms and abbreviations
 * Talk:TLAs: Table of all possible TLAs from AAA to DZZ, to HZZ, to LZZ, to PZZ, to TZZ, to XZZ, to ZZZ
 * Talk:List of proverbial pairs

Physics

 * Talk:List of physics topics
 * Talk:List of equations in classical mechanics
 * Talk:List of laws in science
 * Talk:List of letters used in mathematics and science
 * Talk:List of physics topics
 * Talk:List of satellites which have provided data on the magnetosphere
 * Talk:Physical constant
 * Talk:List of Artificial Radiation Belts
 * Wiktionary: List of colors

Psychology

 * Talk:List of psychological topics
 * Talk:List of emotions
 * Talk:List of phobias

Sociology

 * Talk:List of sociological topics
 * Talk:List of ethnic groups
 * Talk:List of Australian Aboriginal tribes
 * Talk:List of Chinese ethnic groups
 * Talk:List of communities in Nunavut
 * Talk:List of ethnic groups in Laos
 * Talk:List of ethnic groups in Vietnam
 * Talk:List of civic, fraternal, service, and professional organizations
 * Talk:List of literature on rural issues
 * Talk:Types of rural communities

Zoology

 * Talk:List of birds
 * Talk:List of birds of Santa Cruz County, California
 * Talk:Sibley-Monroe checklist 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18
 * Talk:List of cat breeds
 * Talk:List of dog breeds
 * Talk:List of horse breeds
 * Talk:List of historical animals (incl. cats, dogs, horses)

Other

 * Talk:Scientific constants named after people
 * Talk:Scientific laws named after people
 * Talk:Scientific units named after people
 * Talk:List of inventions named after people
 * Talk:List of scientist pairs

Also see

 * Talk:list of academic disciplines
 * Talk:list of articles by category


 * This article is about the province of the Netherlands. For the town in the United States, see Talk:South Holland, Illinois; for the district of Lincolnshire, see Talk:South Holland, England



South Holland (Dutch Zuid-Holland) is a province of the Talk:Netherlands, located in the west of the country on the Talk:North Sea coast. It is one of the most densely populated and industrialized of the provinces. Neighbouring provinces are Talk:Zeeland to the southwest, Talk:Noord Brabant to the southeast, Talk:Gelderland to the east, Utrecht to the northeast and Talk:Noord-Holland to the north.

It contains the major cities of Talk:The Hague (Den Haag or s-Gravenhage) (the seat of government of the country and the seat of the International Court of Justice) and Talk:Rotterdam. Talk:Leiden, Talk:Delft and Talk:Gouda have town centers with many 17th-century buildings. Rotterdam has one of the world's largest harbours.

Rivers and other bodies of water include Talk:Nieuwe Maas, Talk:Nieuwe Waterweg, Talk:Oude Maas, Talk:Haringvliet, Talk:Hollands Diep.

Municipalities
South Holland is divided into 86 municipalities) (before 2004: 91) (here with Talk:shopping evenings in parentheses, and links to maps in brackets):




 * 1) Talk:Alblasserdam
 * 2) Talk:Albrandswaard
 * 3) Talk:Alkemade (Roelofarendsveen: fr)
 * 4) Talk:Alphen aan den Rijn (fr)
 * 5) Talk:Barendrecht
 * 6) Talk:Bergambacht
 * 7) Talk:Bergschenhoek
 * 8) Talk:Berkel en Rodenrijs
 * 9) Talk:Bernisse
 * 10) Talk:Binnenmaas
 * 11) Talk:Bleiswijk
 * 12) Talk:Bodegraven
 * 13) Talk:Boskoop
 * 14) Talk:Brielle
 * 15) Talk:Capelle aan den IJssel
 * 16) Talk:Cromstrijen
 * 17) Talk:Delft (fr)
 * 18) Talk:Den Haag - including Talk:Scheveningen (th)
 * 19) Talk:Dirksland
 * 20) Talk:Dordrecht
 * 21) Talk:Giessenlanden
 * 22) Talk:Goedereede
 * 23) Talk:Gorinchem
 * 24) Talk:Gouda
 * 25) Talk:Graafstroom
 * 26) Talk:'s-Gravendeel
 * 27) Talk:Hardinxveld-Giessendam
 * 28) Talk:Hellevoetsluis
 * 29) Talk:Hendrik-Ido-Ambacht
 * 30) Talk:Hillegom (th)
 * 31) Talk:Jacobswoude
 * 32) Talk:Katwijk (th, except Hoornes and Rijnsoever: fr)
 * 33) Talk:Korendijk
 * 34) Talk:Krimpen aan den IJssel
 * 35) Talk:Leerdam
 * 36) Talk:Leiden (th, except Merenwijk: fr)
 * 37) Talk:Leiderdorp (fr)
 * 38) Talk:Leidschendam-Voorburg
 * 39) Talk:Liemeer
 * 40) Talk:Liesveld
 * 41) Talk:Lisse (th)
 * 42) Talk:Maassluis
 * 43) Talk:Middelharnis
 * 44) Talk:Midden-Delfland (until 2004 Maasland and Schipluiden}
 * 45) Talk:Moordrecht
 * 46) Talk:Nederlek
 * 47) Talk:Nieuw-Lekkerland
 * 48) Talk:Nieuwerkerk aan den IJssel
 * 49) Talk:Nieuwkoop
 * 50) Talk:Noordwijk (th)
 * 51) Talk:Noordwijkerhout (fr)
 * 52) Talk:Oegstgeest (fr)
 * 53) Talk:Oostflakkee
 * 54) Talk:Oud-Beijerland
 * 55) Talk:Ouderkerk
 * 56) Talk:Papendrecht
 * 57) Talk:Pijnacker-Nootdorp (Nootdorp: fr)
 * 58) Talk:Reeuwijk
 * 59) Talk:Ridderkerk
 * 60) Talk:Rijnsburg (fr)
 * 61) Talk:Rijnwoude
 * 62) Talk:Rijswijk
 * 63) Talk:Rotterdam - including Talk:Hoek van Holland (fr)
 * 64) Talk:Rozenburg
 * 65) Talk:Sassenheim (th)
 * 66) Talk:Schiedam
 * 67) Talk:Schoonhoven
 * 68) Talk:Sliedrecht
 * 69) Talk:Spijkenisse
 * 70) Talk:Strijen
 * 71) Talk:Ter Aar
 * 72) Valkenburg (fr)
 * 73) Talk:Vlaardingen
 * 74) Talk:Vlist
 * 75) Talk:Voorhout (fr)
 * 76) Talk:Voorschoten (fr)
 * 77) Talk:Waddinxveen
 * 78) Talk:Warmond
 * 79) Talk:Wassenaar
 * 80) Westland (until 2004 De Lier, 's-Gravenzande, Monster, Naaldwijk and Wateringen)
 * 81) Talk:Westvoorne
 * 82) Talk:Zederik
 * 83) Talk:Zevenhuizen-Moerkapelle
 * 84) Talk:Zoetermeer (th)
 * 85) Talk:Zoeterwoude
 * 86) Zwijndrecht - including Talk:Heerjansdam ,

On 1 January 2004 the municipalities De Lier, 's-Gravenzande, Monster, Naaldwijk and Wateringen have been merged to a new municipality Westland, and Maasland and Schipluiden to another one, Talk:Midden-Delfland.

Possibly Talk:Sassenheim, Talk:Voorhout and Talk:Warmond will be merged in the future.

Links to municipalities arranged according to location
(please add more municipalities)

Talk:Noordwijk Talk:Katwijk Talk:Leiden             Talk:Alphen aan den Rijn Talk:Wassenaar Talk:Voorschoten                         Talk:Bodegraven Talk:Leidschendam-Voorburg Talk:The Hague Talk:Zoetermeer                  Talk:Gouda Talk:Rijswijk Talk:Nootdorp Talk:Delft           Talk:Capelle aan den IJssel Talk:Westland Talk:Vlaardingen Talk:Schiedam Talk:Rotterdam Talk:Barendrecht Talk:Zwijndrecht Talk:Dordrecht Talk:Sliedrecht Talk:Gorinchem

Islands
(from north to south and from west to east, with municipalities)


 * Talk:Rozenburg
 * Rozenburg
 * part of Talk:Rotterdam (Botlek, Talk:Europoort , Maasvlakte  )
 * Talk:IJsselmonde
 * part of Talk:Rotterdam
 * Talk:Albrandswaard
 * Talk:Barendrecht
 * Talk:Ridderkerk
 * Talk:Hendrik-Ido-Ambacht
 * Zwijndrecht ,
 * Talk:Voorne-Putten
 * Talk:Westvoorne
 * Talk:Brielle
 * Talk:Hellevoetsluis
 * Talk:Bernisse
 * Talk:Spijkenisse
 * Talk:Hoeksche Waard (the six municipalities are possibly merged in 2005 or 2006)
 * Talk:Oud-Beijerland
 * Talk:Binnenmaas
 * Talk:'s-Gravendeel
 * part of Talk:Korendijk
 * Talk:Cromstrijen
 * Talk:Strijen
 * Eiland van Talk:Dordrecht
 * Dordrecht
 * Talk:Goeree-Overflakkee
 * Talk:Goedereede
 * Talk:Dirksland
 * Talk:Middelharnis
 * Talk:Oostflakkee.
 * Tiengemeten
 * part of Talk:Korendijk

Subdivisions
There are four official regions :
 * Zuid-Holland Zuid
 * Drechtsteden
 * Alblasserdam, Dordrecht, Hendrik-Ido-Ambacht, Papendrecht, Sliedrecht, Zwijndrecht, 's-Gravendeel
 * the rest of the Hoeksche Waard (see above) and Tiengemeten
 * Goeree Overflakkee (see above)
 * Zuid-Holland West
 * Haaglanden, Talk:w:nl:Stadsgewest Haaglanden;
 * Delft, Den Haag, Leidschendam-Voorburg, Midden-Delfland, Pijnacker-Nootdorp, Rijswijk, Wassenaar, Westland, Zoetermeer.
 * Leidse regio ; for a little map showing the municipalities see, p.4);
 * Alkemade, Leiden, Leiderdorp, Oegstgeest, Voorschoten, Zoeterwoude
 * Duin- en Bollenstreek ; for a little map showing the municipalities see the link above;
 * Hillegom, Katwijk, Lisse, Noordwijk, Noordwijkerhout, Rijnsburg, Sassenheim, Valkenburg, Voorhout, Warmond
 * Zuid-Holland Oost (Rijn&Gouwestreek + De Waarden) ; see also Midden-Holland
 * Krimpenerwaard
 * Bergambacht, Nederlek, Schoonhoven, Vlist, Ouderkerk
 * Alblasserwaard-Vijfheerenlanden
 * Giessenlanden, Gorinchem, Graafstroom, Hardinxveld-Giessendam, Leerdam, Liesveld, Nieuw-Lekkerland, Zederik.
 * Rijnstreek
 * Ter Aar, Alphen aan den Rijn, Jacobswoude, Liemeer, Nieuwkoop, Rijnwoude
 * Gouwestreek
 * Bodegraven, Boskoop, Gouda, Moordrecht, Nieuwerkerk aan den IJssel, Reeuwijk, Waddinxveen, Zevenhuizen-Moerkapelle
 * Rijnmond, also called Stadsregio Rotterdam ; see also Talk:w:nl:Stadsregio Rotterdam
 * Albrandswaard, Barendrecht, Bergschenhoek, Berkel en Rodenrijs, Bernisse, Bleiswijk, Brielle, Capelle aan den IJssel, Hellevoetsluis, Krimpen aan den IJssel, Maassluis, Ridderkerk, Rotterdam, Rozenburg, Schiedam, Spijkenisse, Vlaardingen, Westvoorne

The term Zuidvleugel refers to a large part of the province (,, p. 13):


 * Drechtsteden and some more of the Hoeksche Waard: Binnenmaas, Oud-Beijerland
 * Zuid-Holland West
 * Rijnmond
 * a small part of Zuid-Holland Oost: Gorinchem, Hardinxveld-Giessendam

The necessity for propulsion systems
Artificial satellites must be Talk:launched into Talk:orbit, and once there they must accelerate to circularize their orbit. Once in the desired orbit, they often need some form of Talk:attitude control so that they are correctly pointed with respect to the Talk:Earth, the Talk:Sun, and possibly some astronomical object of interest. They are also subject to drag from the thin atmosphere, so that to stay in orbit for a long period of time some form of propulsion is occasionally necessary to make small corrections (Talk:orbital stationkeeping). Many satellites need to be moved from one orbit to another from time to time, and this also requires propulsion. When a satellite has exhausted its ability to adjust its orbit, its useful life is over.

Spacecraft designed to travel further also need propulsion methods. They need to be launched out of the Earth's atmosphere just as do satellites. Once there, they need to leave orbit and move around.

For Talk:interplanetary travel, a spacecraft must use its engines to leave Earth orbit. Once it has done so, it must somehow make its way to its destination. Current interplanetary spacecraft do this with a series of short-term orbital adjustments. In between these adjustments, the spacecraft simply falls freely along its orbit. The simplest fuel-efficient means to move from one circular orbit to another is with a Talk:Hohmann transfer orbit: the spacecraft begins in a roughly circular orbit around the Sun. A short period of Talk:thrust in the direction of motion accelerates or decelerates the spacecraft into an elliptical orbit around the Sun which is tangential to its previous orbit and also to the orbit of its destination. The spacecraft falls freely along this elliptical orbit until it reaches its destination, where another short period of thrust accelerates or decelerates it to match the orbit of its destination. Special methods such as Talk:aerobraking are sometimes used for this final orbital adjustment.

Some spacecraft propulsion methods such as Talk:solar sails provide very low but inexhaustible thrust; an interplanetary vehicle using one of these methods would follow a rather different trajectory, either constantly thrusting against its direction of motion in order to decrease its distance from the Sun or constantly thrusting along its direction of motion to increase its distance from the Sun.

Spacecraft for Talk:interstellar travel also need propulsion methods. No such spacecraft has yet been built, but many designs have been discussed. Since interstellar distances are very great, a tremendous velocity is needed to get a spacecraft to its destination in a reasonable amount of time. Acquiring such a velocity on launch and getting rid of it on arrival will be a formidable challenge for spacecraft designers.

Effectiveness of propulsion systems
When in space, the purpose of a propulsion system is to change the velocity v of a spacecraft. Since this is more difficult for more massive spacecraft, designers generally discuss Talk:momentum, mv. The amount of change in momentum is called Talk:impulse. So the goal of a propulsion method in space is to create an impulse.

When launching a spacecraft from the Earth, a propulsion method must overcome the Earth's gravitational pull in addition to providing acceleration.

The rate of change of velocity is called Talk:acceleration, and the rate of change of momentum is called Talk:force. To reach a given velocity, one can apply a small acceleration over a long period of time, or one can apply a large acceleration over a short time. Similarly, one can achieve a given impulse with a large force over a short time or a small force over a long time. This means that for maneuvering in space, a propulsion method that produces tiny accelerations but runs for a long time can produce the same impulse as a propulsion method that produces large accelerations for a short time. When launching from a planet, tiny accelerations cannot overcome the planet's gravitational pull and so cannot be used.

The law of Talk:conservation of momentum means that in order for a propulsion method to change the momentum of a space craft it must change the momentum of something else as well. A few designs take advantage of things like magnetic fields or light pressure in order to change the spacecraft's momentum, but in free space the rocket must bring along some mass to accelerate away in order to push itself forward. Such mass is called reaction mass.

In order for a rocket to work, it needs two things: reaction mass and energy. The impulse provided by launching a particle of reaction mass having mass m at velocity v is mv. But this particle has kinetic energy mv2/2, which must come from somewhere. In a conventional solid fuel rocket, the fuel is burned, providing the energy, and the reaction products are allowed to flow out the back, providing the reaction mass. In an Talk:ion thruster, electricity is used to accelerate ions out the back. Here some other source must provide the electrical energy (perhaps a Talk:solar panel or a Talk:nuclear reactor) while the ions provide the reaction mass.

When discussing the efficiency of a propulsion system, designers often focus on the reaction mass. After all, energy can in principle be produced without much difficulty, but the reaction mass must be carried along with the rocket and irretrievably consumed when used. A way of measuring the amount of impulse that can be obtained from a fixed amount of reaction mass is the Talk:specific impulse. This is the impulse per unit mass in newton seconds per kilogram (Ns/kg). This corresponds to metres per second (m/s), and is the effective exhaust velocity ve.

A rocket with a high exhaust velocity can achieve the same impulse with less reaction mass. However, the kinetic energy is proportional to the square of the exhaust velocity, so that more efficient engines (in the sense of having a large specific impulse) require more energy to run.

A second problem is that if the engine is to provide a large amount of thrust, that is, a large amount of impulse per second, it must also provide a large amount of energy per second. So highly efficient engines require enormous amounts of energy per second to produce high thrusts. As a result, most high-efficiency engine designs also provide very low thrust.

Calculations
Burning the entire usable propellent of a spacecraft through the engines in a straight line would produce a net velocity change to the vehicle- this number is termed 'Talk:delta-v'.

The total $$\Delta v$$ of a vehicle can be calculated using the rocket equation, where M is the mass of fuel, P is the mass of the payload (including the rocket structure), and $$I_{sp}$$ is the Talk:specific impulse of the rocket. This is known as the Talk:Tsiolkovsky rocket equation:


 * $$ \Delta V = -I_{sp} \ln \left(\frac{P}{M+P}\right) $$

For a long voyage, the majority of the spacecraft's mass may be reaction mass. Since a rocket must carry all its reaction mass with it, most of the first reaction mass goes towards accelerating reaction mass rather than payload. If we have a payload of mass P, the spacecraft needs to change its velocity by $$\Delta v$$, and the rocket engine has exhaust velocity ve, then the mass M of reaction mass which is needed can be calculated using the rocket equation and the formula for $$I_{sp}$$


 * $$ M = P \left(e^{\Delta v/v_e}-1\right).$$

For $$\Delta v$$ much smaller than ve, this equation is roughly linear, and little reaction mass is needed. If $$\Delta v$$ is comparable to ve, then there needs to be about twice as much fuel as combined payload and structure (which includes engines, fuel tanks, and so on). Beyond this, the growth is exponential; speeds much higher than the exhaust velocity require very high ratios of fuel mass to payload and structural mass.

In order to achieve this, some amount of energy must go into accelerating the reaction mass. Every engine will waste some energy, but even assuming 100% efficiency, the engine will need energy amounting to


 * $$\begin{matrix} \frac{1}{2} \end{matrix} Mv_e^2,$$

This formula reflects the fact that even with 100% engine efficiency, certainly not all energy supplied ends up in the vehicle - some of it, indeed usually most of it, ends up as kinetic energy of the exhaust.

For a mission, for example, when launching from or landing on a planet, the effects of gravitational attraction and any atmospheric drag must be overcome by using fuel. It is typical to combine the effects of these and other effects into an effective mission Talk:delta-v. For example a launch mission to low Earth orbit requires about 9.3-10 km/s delta-v. These mission delta-vs are typically numerically integrated on a computer.

Suppose we want to send a 10,000-kg space probe to Mars. The required $$\Delta v$$ from LEO is approximately 3000 m/s, using a Talk:Hohmann transfer orbit. (A manned probe would need to take a faster route and use more fuel). For the sake of argument, let us say that the following thrusters may be used:

Observe that the more fuel-efficient engines can use far less fuel; its mass is almost negligible (relative to the mass of the payload and the engine itself) for some of the engines. However, note also that these require a large total amount of energy. At one gravity, the total acceleration takes about 300 s, or about five minutes. So, for it to be possible for one of the high-efficiency engines to generate a gravity of thrust, they would have to be supplied with 2.5 or 15 GW of power -  equivalent to a major metropolitan generating station. This would need to be included in the 10,000 kg of payload and structural weight, which is clearly impractical.

Instead, a much smaller, less powerful generator may be included which will take much longer to generate the total energy needed. This lower power is only sufficient to accelerate a tiny amount of fuel per second, but over long periods the velocity will be finally achieved. For example. it took the Talk:Smart 1 more than a year to reach the Moon, while with a chemical rocket it takes a few days. The orbit is not a Hohmann transfer orbit. The launched mass is often lower, which can lower cost.

Interestingly, for a mission delta-v, there is a fixed $$I_{sp}$$ that minimises the overall energy used by the rocket. This comes to an exhaust velocity of about 2/3 of the delta-v (see also the energy computed from the rocket equation). Drives such as VASIMR, and to a lesser extent other Ion thrusters have exhaust velocities that can be enormously higher than this ideal, and thus end up powersource limited and give very low thrust. If the vehicle performance is limited by available power, e.g. if Talk:solar power is used, then in the case of a large $$v_e$$ the possible acceleration is inversely proportional to it, hence the time to reach a required delta-v is proportional to $$v_e$$. Thus the latter should not be too large.

Propulsion methods
Propulsion methods can be classified based on their means of accelerating the reaction mass. There are also some special methods for launches, planetary arrivals, and landings.

Rockets
A rocket engine accelerates its reaction mass by heating it, producing hot high-pressure Talk:gas or Talk:plasma. The reaction mass is then allowed to escape from the rear of the vehicle by passing through a Talk:de Laval nozzle, which dramatically accelerates the reaction mass, converting thermal energy into kinetic energy. It is this nozzle which gives a rocket engine its characteristic shape.

Rockets emitting gases are limited by the fact that their exhaust temperature cannot be so high that the nozzle and reaction chamber are damaged; most large rockets have elaborate cooling systems to prevent damage to either component. Rockets emitting plasma can potentially carry out reactions inside a Talk:magnetic bottle and release the plasma via a Talk:magnetic nozzle, so that no solid matter need come in contact with the plasma. Of course, the machinery to do this is complex, but research into Talk:nuclear fusion has developed methods.



Rocket engines that could be used in space (all emit gases unless otherwise noted):
 * Talk:Solid rocket (chemical energy)
 * Talk:Hybrid rocket (chemical energy)
 * Talk:Monopropellant rocket (chemical energy)
 * Talk:Bipropellant rocket (chemical energy)
 * Talk:Tripropellant rocket (chemical energy)
 * Talk:Dual mode propulsion rocket (chemical energy)
 * Talk:Resistojet rocket (electric heating)
 * Talk:Arcjet rocket (chemical burning aided by electrical discharge)
 * Talk:Pulsed plasma thruster (electric arc heating; emits plasma)
 * Talk:Variable specific impulse magnetoplasma rocket (electromagnetic heating; emits plasma)
 * Talk:Solar thermal rocket (solar energy)
 * Talk:Nuclear thermal rocket (nuclear fission energy)
 * Talk:Radioisotope rocket/Talk:Poodle thruster (nuclear fission energy)
 * Talk:Antimatter catalyzed nuclear pulse propulsion (fission and/or fusion energy)
 * Talk:Gaseous fission reactor (nuclear fission energy)
 * Talk:fission-fragment rocket (nuclear fission energy)
 * Talk:fission sail (nuclear fission energy)
 * Talk:Nuclear salt-water rocket (nuclear fission energy)
 * Talk:Nuclear pulse propulsion (exploding fission/fusion bombs)
 * Talk:Fusion rocket (nuclear fusion energy)
 * Talk:Antimatter rocket (annihilation energy)
 * Talk:Redshift rocket (annihilation energy; hypothetical modification to allow high exhaust temperature without causing damage)

When launching a vehicle from the Earth's surface, the atmosphere poses problems. For example, the precise shape of the most efficient de Laval nozzle for a rocket depends strongly on the ambient pressure. For this reason, various exotic nozzle designs such as the Talk:plug nozzle, the Talk:expanding nozzle and the Talk:aerospike have been proposed, each having some way to adapt to changing ambient air pressure.

On the other hand, rocket engines have been proposed that take advantage of the air in some way (as do Talk:jet engines and other air-breathing engines):
 * the Talk:Air-augmented rocket, and
 * the Talk:Liquid air cycle engine.

Electromagnetic acceleration of reaction mass
Rather than relying on high temperature and Talk:fluid dynamics to accelerate the reaction mass to high speeds, there are a variety of methods that use electrostatic or electromagnetic forces to accelerate the reaction mass directly. Usually the reaction mass is a stream of Talk:ions. Such an engine requires electric power to run, and high exhaust velocities require large amounts of power to run.

It turns out that to a reasonable approximation, for these drives, that fuel use, energetic efficiency and thrust are all inversely proportional to exhaust velocity. Their very high exhaust velocity means they require huge amounts of energy and provide low thrust; but use hardly any fuel.

For some missions, Talk:solar energy may be sufficient, but for others nuclear energy will be necessary; engines drawing their power from a nuclear source are called Talk:nuclear electric rockets. With any current source of power, the maximum amount of power that can be generated limits the maximum amount of thrust that can be produced while adding significant mass to the spacecraft.

Some electromagnetic methods:
 * Talk:Hall effect thruster
 * Talk:Ion thruster
 * Talk:Field Emission Electric Propulsion
 * Talk:Magnetoplasmadynamic thruster
 * Talk:Pulsed inductive thruster
 * Talk:Variable specific impulse magnetoplasma rocket
 * Talk:Mass drivers (for propulsion)

The Talk:Biefeld-Brown effect is a somewhat exotic electrical effect. In Talk:air, a voltage applied across a particular kind of Talk:capacitor produces a thrust. There have been claims that this also happens in a Talk:vacuum due to some sort of coupling between the Talk:electromagnetic field and Talk:gravity, but recent experiments show no evidence of this hypothesis.

Systems without reaction mass
The law of conservation of Talk:momentum states that any engine which uses no reaction mass cannot move the center of mass of a spaceship (changing orientation, on the other hand, is possible). But space is not empty, especially space inside the Solar Systems; there is a Talk:magnetic field and a Talk:solar wind. Various propulsion methods try to take advantage of this; since all these things are very diffuse, propulsion structures need to be large.

Space drives that need no (or little) reaction mass:
 * Talk:Momentum wheel
 * Talk:Tether propulsion
 * Talk:Solar sails
 * Talk:Magnetic sails
 * Talk:Mini-magnetospheric plasma propulsion

Launch mechanisms
High thrust is of vital importance for launch, the thrust per unit mass has to be well above g, see also Talk:gravity drag. Many of the propulsion methods above do not provide that much thrust, especially if solar power is used. In that case, at the very least the mass of the solar panel would have to be less than 20 gram per 1000 W of power, and even less if the specific impulse is higher or lower than the optimum value, which would be in the order of magnitude of 10 km/s; also the engine would have to be very light and energy-efficient.

Exhaust toxicity or other side effects can also have detrimental effects on the environment the spacecraft is launching from, ruling out other propulsion methods.

Therefore, all current spacecraft use chemical rocket engines (bipropellant or solid-fuel) for launch.

One advantage that spacecraft have in launch is the availability of infrastructure on the ground to assist them. Proposed ground-assisted launch mechanisms include:


 * Talk:Space elevator
 * Talk:Orbital airship
 * Talk:Space fountain
 * Talk:Hypersonic skyhook
 * Electromagnetic catapult (Talk:rail gun, Talk:coil gun)
 * Ballistic acceleration (Talk:Project HARP, Talk:ram accelerator)
 * Talk:Laser propulsion (Talk:Lightcraft)

Planetary arrival and landing
When a vehicle is to enter orbit around its destination planet, or when it is to land, it must adjust its velocity. This can be done using all the methods listed above (provided they can generate a high enough thrust), but there are a few methods that can take advantage of planetary atmospheres.


 * Talk:aerobraking brings a probe into orbit
 * Talk:parachutes can land a probe on a planet with an atmosphere
 * Talk:airbags can soften the final landing

Talk:Gravitational slingshots can also be used to carry a probe onward to other destinations.

Methods requiring new principles of physics
In addition, a variety of hypothetical propulsion techniques have been considered that would require entirely new principles of physics to realize. As such, they are currently highly speculative:


 * Talk:Alcubierre drive (Warp drive)
 * Talk:Wormholes
 * Talk:Differential sail
 * Talk:Disjunction drive
 * Talk:Diametric drive
 * Talk:Pitch drive
 * Talk:Bias drive
 * Talk:Time machines

Table of methods and their specific impulse
Below is a summary of some of the more popular, proven technologies, followed by increasingly speculative methods.

Three numbers are shown. The first is the Talk:specific impulse: the amount of thrust that can be produced using a unit of fuel. This is the most important characteristic of the propulsion method:
 * if the delta-v is much more, then exorbitant amounts of fuel are necessary (see the section on calculations, above)
 * if it is much more than the delta-v, then, proportionally more energy is needed; if the power is limited, as with solar energy, this means that the journey takes a proportionally longer time

The second and third are the typical amounts of thrust and the typical burn times of the method. Outside a gravitational potential small amounts of thrust applied over a long period will give the same effect as large amounts of thrust over a short period.

This result does not apply when the object is influenced by gravity.