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Solar Day

Solar day

Solar time is based on the idea that when the sun reaches its highest point in the sky, it is noon. Apparent solar time is based on the apparent solar day, which is the interval between two successive returns of the Sun to the local meridian. Solar time can be measured by a sundial. The length of a solar day varies throughout the year for two reasons. First, the Earth's orbit is an ellipse, not a circle, so the Earth moves faster when it is nearest the Sun and slower when it is farthest from the Sun (see Kepler's laws of planetary motion). Second, due to Earth's axial tilt, the Sun does not usually appear to move along Earth's celestial equator but usually appears to move at an angle to it during the year; thus, the sun appears to move either fast or slow depending on whether it appears to be far from or close to the equator (see tropical year). Consequently, apparent solar days are shorter in March (26–27) and September (12–13) than they are in June (18–19) or December (20–21). Mean solar time is artificial clock time adjusted via observations of the diurnal rotation of the fixed stars to agree with average apparent solar time. The length of a mean solar day is a constant 24 hours throughout the year even though the amount of daylight within it may vary. An apparent solar day may differ from a mean solar day (of 86,400 seconds) by as much as nearly 22 seconds shorter to nearly 29 seconds longer. Because many of these long or short days occur in succession, the difference builds up to as much as nearly 17 minutes early or a little over 14 minutes late. The difference between apparent solar time and mean solar time is called the equation of time. Many methods have been used to simulate mean solar time throughout history. The earliest were clepsydras or water clocks, used for almost four millennia from as early as the middle of the second millennium BC until the early second millennium. Before the middle of the first millennium BC, the water clocks were only adjusted to agree with the apparent solar day, thus were no better than the shadow cast by a gnomon (a vertical pole), except that they could be used at night. Nevertheless, it has always been known that the sun moves eastward relative to the fixed stars along the ecliptic. Thus since the middle of the first millennium BC, the diurnal rotation of the fixed stars has been used to determine mean solar time, against which clocks were compared to determine their error rate. Babylonian astronomers knew of the equation of time and were correcting for it as well as the different rotation rate of stars, sidereal time, to obtain a mean solar time much more accurate than their water clocks. This ideal mean solar time has been used ever since then to describe the motions of the planets, Moon, and Sun. Mechanical clocks did not achieve the accuracy of Earth's 'star clock' until the beginning of the twentieth century. Even though today's atomic clocks have a much more constant rate than the Earth, its star clock is still used to determine mean solar time. Since sometime in the late 1900s, Earth's rotation has been defined relative to an ensemble of extra-galactic radio sources and then converted to mean solar time by an adopted ratio. The difference between this calculated mean solar time and Coordinated Universal Time (UTC) is used to determine whether a leap second is needed.

Sun

:: For the astrological significance of the Sun, see Solar system in astrology. ::"Solar" redirects here; for the superhero by that name, see Solar (comics). The Sun (or Sol) is the star at the center of our Solar system. Earth orbits the Sun, as do many other bodies, including other planets, asteroids, meteoroids, comets and dust. Its heat and light support almost all life on Earth. The Sun is a ball of plasma with a mass of about 2 kg, which is somewhat higher than that of an average star. About 74% of its mass is hydrogen, with 25% helium and the rest made up of trace quantities of heavier elements. It is thought that the Sun is about 5 billion years old, and is about halfway through its main sequence evolution, during which nuclear fusion reactions in its core fuse hydrogen into helium. In about 5 billion years time the Sun will become a white dwarf. Although it is the nearest star to Earth and has been intensively studied by scientists, many questions about the Sun remain unanswered, such as why its outer atmosphere has a temperature of over 10⁶ K when its visible surface (the photosphere) has a temperature of just 6,000 K. Looking directly at the Sun can damage the retina and one's eyesight. See below for details.

General information

See below The Sun is classified as a main sequence star, which means it is in a state of "hydrostatic balance", neither contracting nor expanding, and is generating its energy through nuclear fusion of hydrogen nuclei into helium. The Sun has a spectral class of G2V, with the G2 meaning that its color is yellow and its spectrum contains spectral lines of ionized and neutral metals as well as very weak hydrogen lines [http://www.astro.uiuc.edu/~kaler/sow/spectra.html#classes], and the V signifying that it, like most stars, is a "dwarf" star on the main sequence[http://www.physics.uq.edu.au/people/ross/phys2080/spec/analyz.htm]. The Sun has a predicted main sequence lifetime of about 10 billion years. Its current age is thought to be about 4.5 billion years, a figure which is determined using computer models of stellar evolution, and nucleocosmochronology . The Sun orbits the center of the Milky Way galaxy at a distance of about 25,000 to 28,000 light-years from the galactic centre, completing one revolution in about 226 million years. The orbital speed is 217 km/s, equivalent to one light year every 1400 years, and one AU every 8 days. The astronomical symbol for the Sun is a circle with a point at its centre (Image:Sol.gif).

Structure

Image:Sol.gif The Sun is a near-perfect sphere, with an oblateness estimated at about 9 millionths, which means the polar diameter differs from the equatorial by about 10 km. This is because the centrifugal effect of the Sun's slow rotation is 18 million times weaker than its surface gravity (at the equator). Tidal effects from the planets do not significantly affect the shape of the Sun, although the Sun itself orbits the center of mass of the solar system, which is offset from the Sun's center mostly because of the large mass of Jupiter. The mass of the Sun is so comparatively great that the center of mass of the solar system is generally within the bounds of the Sun itself. The Sun does not have a definite boundary as rocky planets do, as the density of its gases drops off following an approximately exponential relationship with distance from the centre of the Sun. Nevertheless, the Sun has well defined interior structure, described below. The Sun's radius is measured from centre to the edges of the photosphere. The solar interior is not directly observable and the Sun itself is opaque to electromagnetic radiation. However, just as the study of the waves generated by earthquakes (seismology) can be used to study the interior structure of the Earth, helioseismology, the study of sound waves that travel through the Sun's interior, has also contributed greatly to our understanding of the Sun's structure . Computer modeling of the Sun is also used as a theoretical tool to investigate its deep layers.

Core

At the center of the Sun, where its density reaches up to 150,000 kg/m³ (150 times the density of water on Earth), thermonuclear reactions (nuclear fusion) convert hydrogen into helium, producing the energy that keeps the Sun in a state of equilibrium. About 8.9 protons (hydrogen nuclei) are converted to helium nuclei every second, releasing energy at the matter-energy conversion rate of 4.26 million tonnes per second or 383 yottawatts (9.15 tons of TNT per second). The core extends from the center of the Sun to about 0.2 solar radii, and is the only part of the Sun where an appreciable amount of heat is produced by fusion: the rest of the star is heated by energy that is transferred outward. All of the energy of the interior fusion must travel through the successive layers to the solar photosphere, before it escapes to space. The high-energy photons (gamma and X rays) released in fusion reactions take a long time to reach the Sun's surface, slowed down by the indirect path taken, as well as constant absorption and re-emission at lower energies in the solar mantle (see below). Estimates of the "photon travel time" range from as much as 50 million years (Richard S. Lewis, The Illustrated Encyclopedia of the Universe, Harmony Books, New York, 1983, p. 65) to as little as 17,000 years [http://www.badastronomy.com/bitesize/solar_system/sun.html]. Upon reaching the surface after a final trip through the convective outer layer, the photons escape as visible light. Neutrinos are also released in the fusion reactions in the core, but unlike photons they very rarely interact with matter, and so almost all are able to escape the Sun immediately.

Radiation zone

From about 0.2 to about 0.7 solar radii, the material is hot and dense enough that thermal radiation is sufficient to transfer the intense heat of the core outward. In this zone, there is no thermal convection: while the material grows cooler with altitude, this temperature gradient is slower than the adiabatic lapse rate and hence cannot drive convection. Heat is transferred by ions of hydrogen and helium emitting photons, which travel a brief distance before being re-absorbed by other ions. Because of this, it can take a photon nearly 1,000,000 years to reach the photosphere.

Convection zone

photosphere From about 0.7 solar radii to 1.0 solar radii, the material in the Sun is not dense enough or hot enough to transfer the heat energy of the interior outward via radiation. As a result, thermal convection occurs as thermal columns carry hot material to the surface (photosphere) of the Sun. Once the material cools off at the surface, it plunges back downward to the base of the convection zone, to receive more heat from the top of the radiative zone. Convective overshoot is thought to occur at the base of the convection zone, carrying turbulent downflows into the outer layers of the radiative zone. The thermal columns in the convection zone form an imprint on the surface of the Sun, in the form of the solar granulation and supergranulation. The turbulent convection of this outer part of the solar interior gives rise to a 'small-scale' dynamo that produces magnetic north and south poles all over the surface of the Sun.

Photosphere

The visible surface of the Sun, the photosphere, is the layer below which the Sun becomes opaque to visible light. Above the photosphere, sunlight is free to propagate into space and its energy escapes the Sun entirely. Sunlight has approximately a black-body spectrum that indicates its temperature is about 6,000 K, interspersed with atomic absorption lines from the tenuous layers above the photosphere. The photosphere has a particle density of about 10²³/m³ (this is about 1% of the particle density of Earth's atmosphere at sea level). The parts of the Sun above the photosphere are referred to collectively as the solar atmosphere. They can be viewed with telescopes operating across the electromagnetic spectrum, from radio through visible light to gamma rays.

Temperature minimum

The coolest layer of the Sun is the temperature minimum region about 500 km above the photosphere. It is about 4,000 K. It is the only part of the Sun cool enough to support simple molecules such as carbon monoxide and water; all other parts of the Sun are hot enough to break chemical bonds.

Chromosphere

Above the visible surface of the Sun is a thin layer, about 2,000 km thick, that is dominated by a spectrum of emission and absorption lines. It is called the chromosphere from the Greek root chromos, meaning color, because the chromosphere is visible as a colored flash at the beginning and end of total eclipses of the Sun.

Corona

The corona is the extended outer atmosphere of the Sun, which is much larger in volume than the Sun itself. The corona merges smoothly with the solar wind that fills the solar system and heliosphere. The low corona, which is very near the surface of the Sun, has a particle density of 10¹¹/m³ (Earth's atmosphere near sea level has a particle density of about 2x10²⁵/m³). The temperature of the corona is several megakelvins.

Theoretical problems

Solar neutrino problem

megakelvin For some time it was thought that the number of neutrinos produced by the nuclear reactions in the Sun was only a third of the number predicted by theory, a result that was termed the solar neutrino problem. Several neutrino observatories were constructed, including the Sudbury Neutrino Observatory and Kamiokande to try to measure the solar neutrino flux. It has recently been found that neutrinos have rest mass, and can therefore transform into harder-to-detect varieties of neutrinos while en route from the Sun to Earth in a process known as neutrino oscillation . Thus, measurement and theory have been reconciled.

Coronal heating problem

The optical surface of the Sun (the photosphere) is known to have a temperature of about 6,000 K. Above it lies the solar corona with a temperature of one million kelvins. The high temperature of the corona suggests that it is heated by something other than the photosphere. It is thought that the energy necessary to heat the corona is provided by turbulent motion in the convection zone below the photosphere. Two main mechanisms have been proposed to explain coronal heating: Wave heating, in which sound, gravitational and magnetohydrodynamic waves are produced by turbulence in the convection zone. These waves travel upward and dissipate in the corona, depositing their energy in the ambient gas in the form of heat. The other proposed mechanism is flare heating, in which magnetic energy is continuously built up by photospheric motion and released through magnetic reconnection in the form of solar flares and waves. , , , . Currently, it is unclear whether waves are an efficient heating mechanism. All waves except Alfven waves have been found to dissipate or refract before reaching the corona (, ). In addition, Alfven waves do not easily dissipate in the corona . Current research focus has therefore shifted towards flare heating mechanisms. One possible candidate to explain coronal heating is continuous flaring at small scales , but this is still an open topic of investigation.

Faint young sun problem

Theoretical models of the sun's development suggest that 3.8 to 2.5 billion years ago, during the Archean period, the Sun was only about 75 percent as bright as it is today. Such a weak star would not have been able to sustain liquid water on the Earth's surface, and thus life should not have been able to develop. However, the geologic record shows that the Earth has remained at a fairly constant temperature throughout its history. In fact, the young Earth was actually warmer than it is today. Some scientists have suggested that the young Earth's atmosphere contained much larger quantities of greenhouse gases such as carbon dioxide and/or ammonia than are present today . Others suggest that cosmic rays might strongly influence the Earth's climate, and that their flux was much higher in the early history of the solar system .

Magnetic field

cosmic ray's rotating magnetic field on the plasma in the interplanetary medium (Solar Wind) [http://quake.stanford.edu/~wso/gifs/HCS.html]. (click to enlarge)]] All matter in the Sun is in the form of gas and plasma due to its high temperatures. This makes it possible for the Sun to rotate faster at its equator (about 25 days) than it does at higher latitudes (28 days near its poles). The differential rotation of the Sun's latitudes causes its magnetic field lines to become twisted together over time, causing magnetic field loops to erupt from the Sun's surface and trigger the formation of the Sun's dramatic sunspots and solar prominences. (See magnetic reconnection.) The solar activity cycle includes old magnetic fields being stripped off the Sun's surface starting from one pole and ending at the other. The magnetic field of the sun reverses once for each 11-year sunspot cycle. The influence of the Sun's rotating magnetic field on the plasma in the interplanetary medium creates the largest structure in the Solar System, the Heliospheric current sheet. The plasma in the interplanetary medium is also responsible for the strength of the Sun's magnetic field at the orbit of the Earth being over 100 times greater than originally anticipated. If space were a vacuum, then the Sun's 10^-4 tesla magnetic dipole field would reduce with the cube of the distance to about 10^-11 tesla. But satellite observations show that it is about 100 times greater at around 10^-9 tesla. Magnetohydrodynamic (MHD) theory predicts that the motion of a conducting fluid (e.g. the interplanetary medium) in a magnetic field, induces electric currents which in turn generates magnetic fields, and in this respect it behaves like an MHD dynamo.

Position of the Sun through the year

The path of the Sun across the sky varies throughout the year. The shape described by the Sun's position, considered at the same time each day for a complete year, is called the analemma, and resembles a figure 8, aligned along the North/South direction. The most obvious variation in the Sun's apparent position through the year is a North/South swing over 47 degrees of angle, due to the 23.5 degree tilt of the Earth, but there is an East/West component as well. The North/South swing in apparent angle is the main source of seasons on Earth.

Solar space missions

seasons using UV light from the He⁺ emission line at 30.4 nm. (Animation (980 kB MPEG))]] To obtain an uninterrupted view of the Sun, the European Space Agency and NASA cooperatively launched the Solar and Heliospheric Observatory (SOHO) on December 2, 1995. Originally a two-year mission, SOHO is now over ten years old (as of late 2005). It has proved so useful that a follow-on mission, the Solar Dynamics Observatory, is planned for launch in 2008. Elemental abundances in the photosphere are well known from spectroscopic studies, but the composition of the interior of the Sun is much less well known. A solar wind sample return mission, Genesis, was designed to allow astronomers to directly measure the composition of solar material. It returned to Earth in 2004 and is undergoing analysis, but it was damaged by crash-landing when its parachute failed to deploy on reentry to Earth's atmosphere.

History and future of the Sun

The Sun is thought to be a second-generation star, whose formation may have been triggered by shockwaves from a nearby supernova. This is suggested by a high abundance of heavy elements such as iron, gold and uranium in the solar system: the most plausible ways that these elements could be produced are by endothermic nuclear reactions during a supernova or by transmutation via neutron absorption inside a massive first generation star. Our Sun does not have enough mass to explode as a supernova, and its mass is below the Chandrasekhar limit. Instead, in 4-5 billion years it will enter its red giant phase, its outer layers expanding as the hydrogen fuel in the core is consumed and the core contracts and heats up. Helium fusion will begin when the core temperature reaches about 3 K. While it is likely that the expansion of the outer layers of the Sun will reach the current position of Earth's orbit, recent research suggests that mass lost from the Sun earlier in its red giant phase will cause the Earth's orbit to move further out, preventing it from being engulfed. Following the red giant phase, giant thermal pulsations will cause the Sun to throw off its outer layers forming a planetary nebula. The Sun will then evolve into a white dwarf, slowly cooling over eons. This stellar evolution scenario is typical of low to medium mass stars.

Human understanding of the Sun

:see also sun worship sun worship mythology]] Mankind's most fundamental understanding of the Sun is as the luminous disk in the heavens whose presence above the horizon creates day, and whose absence causes night. In many prehistoric and ancient cultures, the Sun was thought to be a deity or other supernatural phenomenon. One of the first people in the Western world to offer a scientific explanation for the sun was the Greek philosopher Anaxagoras, who reasoned that it was a giant flaming ball of metal even larger than the Peleponessus, and not the chariot of Helios. For teaching this heresy he was imprisoned by the authorities and sentenced to death (though later released through the intervention of Pericles). With respect to the fixed stars, the Sun appears from Earth to revolve once a year along the ecliptic through the zodiac. Thus, the Sun was considered by Greek astronomers to be one of the seven planets (Greek planetes "wanderer"), after which the seven days of the week are named in some languages.

The Sun as a power source

Sunlight — that is, light radiated from the surface of the Sun — is thought to be the main source of energy near the surface of Earth. The solar constant is the amount of power that the Sun deposits per unit area that is directly exposed to sunlight. It is about 1370 watts per square meter of area. Sunlight on the surface of Earth is attenuated by the Earth's atmosphere, so that less power arrives at the surface — closer to 1000 watts per directly exposed square meter in clear conditions. This energy can be harnessed through several natural and synthetic processes. Photosynthesis by plants captures the energy of sunlight and converts it to chemical form (oxygen and reduced carbon compounds), while direct heating or electrical conversion by solar cells are used by solar power equipment to generate electricity or do other useful work. The energy stored in petroleum is thought to have been converted from sunlight by photosynthesis in the distant past.

Sun and eye damage

Sunlight is very bright, and looking directly at the Sun is painful to the eyes. Looking directly at the Sun when it is high in the sky causes temporary bleaching of the photosensitive pigments in the retina, which makes phosphene visual artifacts and may cause temporary partial blindness. Direct viewing of the Sun with the naked eye delivers about 4 milliwatts of sunlight to the retina that is in the solar image, heating it up and potentially (though not normally) damaging it. Brief viewing of the full direct Sun with the naked eye is unpleasant but generally safe. Viewing the Sun through light-concentrating optics such as binoculars is hazardous without an attenuating (ND) filter to dim the sunlight. Suitable filters are available at welding supply shops and camera stores. Using a proper filter is very important as some improvised filters reduce visible light while passing either infrared or ultraviolet rays that can still damage the eye. Viewing the Sun through unfiltered 7x50 mm binoculars can deliver as much as 2.5 watts of sunlight into each eye, over 300 times more power than naked eye viewing. Even brief glances at the midday Sun through unfiltered binoculars can cause permanent blindness. During partial eclipses of the Sun, another hazardous condition exists because of the way the eye responds to bright light. The pupil is controlled by the total amount of light in the visual field, not by the brightest object in the field. During partial eclipses, most sunlight is blocked by the Moon passing directly in front of the Sun, but the uncovered parts of the photosphere have the same surface brightness as during a normal day. In the dim overall light, the pupil tends to dilate from about 2 mm to perhaps 6 mm diameter, increasing the eye's collecting area by a factor of nearly 10. Each retinal cell that is exposed to the partially-eclipsed solar image thus receives about ten times as much light as it would looking at the normal, non-eclipsed Sun. Viewing the partially eclipsed Sun with the naked eye can cause permanent localized damage to the retina, resulting in small, permanent blind spots for the viewer. This is an especially insidious hazard for inexperienced observers and for children, because there is no immediate perception of pain and it is tempting to stare at the spectacle of the eclipsing Sun, compounding any damage. During sunrise and sunset, sunlight is attenuated by a particularly long passage through Earth's atmosphere, and the direct Sun is sometimes faint enough to be viewed directly without discomfort or safely with binoculars. Hazy conditions, atmospheric dust, and high humidity contribute to this atmospheric attenuation.

External links

- [http://sohowww.nascom.nasa.gov/data/realtime-images.html Current SOHO snapshots]
- [http://soi.stanford.edu/data/farside/index.html Far-Side Helioseismic Holography] from [http://www.stanford.edu Stanford]
- [http://sunearth.gsfc.nasa.gov/eclipse/eclipse.html NASA Eclipse homepage]
- [http://sohowww.nascom.nasa.gov/ Nasa SOHO (Solar & Heliospheric Observatory) satellite] [http://sohowww.nascom.nasa.gov/explore/faq/sun.html FAQ]
- [http://soi.stanford.edu/results/sounds.html Solar Sounds] from [http://www.stanford.edu Stanford]
- [http://www.spaceweather.com Spaceweather.com]
- [http://scienceworld.wolfram.com/astronomy/Sun.html Eric Weisstein's World of Astronomy - Sun]
- [http://www.astro.uu.nl/~strous/AA/en/antwoorden/zonpositie.html The Position of the Sun]
- [http://www.lmsal.com/YPOP/FilmFestival/index.html A collection of solar movies]
- [http://www.solarphysics.kva.se/ The Institute for Solar Physics- Movies of Sunspots and spicules]
- [http://science.msfc.nasa.gov/ssl/pad/solar/default.htm NASA/Marshall Solar Physics website]
- [http://rredc.nrel.gov/solar/codesandalgorithms/spa Solar Position Algorithm] and [http://www.nrel.gov/docs/fy04osti/34302.pdf documentation] from the [http://www.nrel.gov National Renewable Energy Laboratory]
- [http://libnova.sourceforge.net/index.html libnova] - a celestial mechanics and astronomical calculation library

References

# Alfven, H., 1947, Monthly Notices of the Royal Astronomical Society., 107, 211 # # Biermann, L., 1946, Naturwissenschaffen, 33, 118 # Bonanno, A., Schlattl, H., Paternò, L. (2002), The age of the Sun and the relativistic corrections in the EOS, Astronomy and Astrophysics, v.390, p.1115-1118 # Carslaw, K.S., Harrison, R.G., Kirkby, J., 2002, Cosmic Rays, Clouds, and Climate, Science, 298, 1732-1737 # Kasting, J.F., Ackerman, T.P., 1986, Climatic Consequences of Very High Carbon Dioxide Levels in the Earth’s Early Atmosphere, Science, v. 234, p. 1383-1385 # Parker, E.N., 1958, Astrophysical Journal, 128, 644 # Parker, E.N., 1988, Astrophysical Journal, 330, 474 # Priest, E.R., 1982, Solar Magnetohydrodynamics (Dordrecht: Reidel), pp. 206-245 # Schlattl, H. (2001), Three-flavor oscillation solutions for the solar neutrino problem, Physical Review D, vol. 64, Issue 1 # Sturrock, P.A., & Uchida, Y., 1981, Astrophysical Journal., 246, 331 # Thompson, M.J. (2004), Solar interior: Helioseismology and the Sun's interior, Astronomy & Geophysics, v. 45, p. 4.21-4.25 Category:Yellow dwarfs Category:Space plasmas Category:Plasma physics als:Sonne zh-min-nan:Ji̍t-thâu ko:태양 ms:Matahari ja:太陽 simple:Sun th:ดวงอาทิตย์

Meridian

Meridian is:
- Meridian (astronomy): an imaginary circle perpendicular to the horizon.
- Meridian (Chinese medicine): an important concept in traditional Chinese medicine.
- Meridian (geography): either half or a full great circle that connects the Earth's poles.
- Meridian (perimetry, visual field): a polar coordiate system for the visual field.
- The name of some places in the United States of America:
- Meridian, Idaho
- Meridian, Mississippi
- Meridian, New York
- Meridian, Oklahoma
- Meridian is also a novel by Alice Walker.
- Meridian Broadcasting has been the ITV station for the south of England since 1992.
- Meridian Energy Limited is a New Zealand electricity generation and electricity retailing company.
- Meridian was a comic book series published by CrossGen in the early 2000s, and was the name of a city-state in that comic.
- Meridian is the name of a line of business telecommunication products produced and marketed by Nortel Networks.
- Meridian is also a term used by the Public Land Survey System in the United States.
- Meridian is the Capital of Nosgoth in the Legacy of Kain series of video games.
- DJ Meridian is a Drum and Bass musician in New York.
- Meridian 59, an online game.
- The British Rail Class 222 trains are known as Meridians and the the services they are used on are named Midland Mainline Meridian services.

Sundial

:This article pertains to the astronomical instrument. For the psychedelic rock band, see Sun Dial. Sun Dial Sun Dial A sundial measures time by the position of the sun. The most commonly seen designs, such as the 'ordinary' or standard garden sundial, cast a shadow on a flat surface marked with the hours of the day. As the position of the sun changes, the time indicated by the shadow changes. However, sundials can be designed for any surface where a fixed object casts a predictable shadow. Most sundial designs indicate apparent solar time. Minor design variations can measure standard and daylight saving time, as well. Sundials are known from ancient Egypt, and were developed further by other cultures, including the Greeks and Romans. The mathematician and astronomer Theodosius of Bithynia (ca. 160 BC-ca. 100 BC) is said to have invented a universal sundial that could be used anywhere on Earth. The French astronomer Oronce Fine constructed a sundial of ivory in 1524. The Italian astronomer Giovanni Padovani published a treatise on the sundial in 1570, in which he included instructions for the manufacture and laying out of mural (vertical) and horizontal sundials.

Installation of standard sundials

1570 Tilting the style or gnomon of a standard sundial is the only practical way to install a mass-produced garden sundial so that it will keep time. Some mass-produced garden sundials are improperly designed, and unable to keep time. Many sundials are made to be used at 45 degrees north. A sundial can be adjusted to another latitude by tilting it so its style or gnomon(s) is (are) parallel the Earth's axis of rotation. That is, the end of a gnomon should point at the north celestial pole in the northern hemisphere, or the south celestial pole in the southern hemisphere. A sundial can be rotated around its style or gnomon (which must still point at the celestial pole) a maximum of 7.5 degrees to the east or west to adjust to the local standard time zone (time zones are 360 degrees/24 hours = 15 degrees wide). Tilt the sundial so that it is oriented as if it were at the longitude of the center of your local time zone. To correct for daylight saving time, a face needs two sets of numerals or a correction table, and must be adjusted for longitude from the center of the time zone. The admittedly informal standard is to have numerals in hot colors for summer, and in cool colors for winter. Twisting the face of the sundial will not work because sundials (except at the north and south pole) do not have equal hour angles. Ordinary sundials do not correct apparent solar time to clock time. There is a 15 minute variation through the year, known as the equation of time, because the Earth's orbit is slightly elliptical and its axis is tilted relative to the plane of its orbit. A quality sundial will include a permanently-mounted table or graph giving this correction for at least each month of the year. Some more-complex sundials have curved hour-lines, curved gnomons or other arrangements to directly display the clock time.

Design & principles of operation

Terminology

The 'shadow-maker' of the sundial is called a gnomon. The sun casts a shadow from the gnomon to a surface called the dial face or dial plate (often shortened to face). Most sundials indicate time on the dial face by the shadow of a line in space called the style[http://www.sundialsoc.org.uk/glossary/alpha.htm#S]. On a standard garden sundial, this line is the top edge of the gnomon. The style should be parallel the Earth's axis of rotation. In common speech, sometimes style refers to the entire gnomon. Some sundials indicate both the time and the date by the shadow of a particular point on the gnomon. That point is called the nodus. The nodus may be the tip of a gnomon with an arbitrary (usually horizontal or vertical) orientation[http://www.sundialsoc.org.uk/glossary/alpha.htm#N]. A few sundials have both a style and a nodus, with the nodus in the form of a small sphere or a notch on a polar-pointing gnomon, or simply the tip of the gnomon. In general, the best material for a face is a very light color to give a high contrast with the shadow. The numerals should be dark, visible on the unshaded portion of the face. The gnomon should be sturdy, preferably metal, because gnomons are usually thin, and can break easily. The traditional luxury materials are a white marble face, with markings inlaid in black marble. Traditional styles are thick bronze to prevent corrosion. It is traditional for a sundial to have a motto.

Equatorial or Equinoctial[http://www.sundials.co.uk/tbequ.htm] sundial

left The simplest sundial is a disk mounted on a bar. The bar must be parallel to the Earth's axis of rotation. The disk forms a plane parallel to the plane of the Earth's equator. The disk is marked so that one edge of the shadow of the bar shows the time as the Earth rotates. Usually noon will be at the bottom of the disk, 6AM on the western edge, and 6PM on the eastern edge. In the winter, the north side of the disk will be shaded, and hard to read. In the summer, the south side will be shaded. In the above design, the bar is the style. The disk in the above design is called the face. In the summer, the north end of the bar is the nodus, but in the winter, the south end of the bar is the nodus. left A series of cocentric circles can be drawn on the face which plot the path of the shadow of the nodus on specific days, thus the dial can be used as a calendar as well as clock. The style shows the time and the nodus the date. One disadvantage of this design is that with a solid face, near the equinox, when sun is just on the celestial equator, the dial is hard to read.

Garden sundial

equinox The classic garden sundial uses the same principle, except the lines of the disk are projected, using trigonometry, onto a face that is parallel to the ground. The advantage of the garden sundial is that it keeps time all year, and its face is never completely shaded in the daytime (as vertical sundials are). For use in a public area, this sundial can be made visible by placing it in a square, or making the face of frosted glass, elevated high in the air, and visible from underneath. The top edge of the gnomon is parallel with the axis of the Earth's rotation. The shadow will cross time markings on the face.The markings of each edge are aligned with the edge of the gnomon that produces the shadow. The angle of the face markings from the root of the gnomon (the substyle) are calculated from the formula face-angle = arctan(sin(latitude)
- tan(hour-angle)). The angle of the style (gnomon)= 90 - latitude. (See Logo programming language for a sample program to draw a garden sundial)

Vertical sundials

Logo programming language Logo programming language Although they are rare in modern life, sundials on vertical south-facing walls (north-facing in the southern hemisphere) are a traditional ancient convenience. They are easy to see from large distances and inexpensive to arrange. One sturdy method is to paint the sundial on the wall, and construct the gnomon as a tripod of metal bars. Fancy sundials used to have faces of inlaid stone. A problem is that vertical sundials only keep time for the part of the year in which the sun illuminates the wall. They are very similar to garden sundials. The formula for a south-facing sundial face is face-angle = arctan(cos(latitude)
- tan(hour-angle)). The angle of the style (gnomon)= latitude. It used to be traditional to place four sundials on the roof or sides of a tower to provide the time. In this way, the time was available to all for the entire year. In principle, sundials can be placed on any surface, at any angle, given the correct trigonometric projection of the face. For example, sundials on roofs are harder to calculate but quite practical.

Portable sundials, for navigation and time

During the middle ages advanced yet portable astronomical instruments were developed.

Diptych sundial

One popular portable sundial design was called a diptych. It consisted of two small flat faces, joined by a hinge. Diptychs usually folded into little flat boxes suitable for a pocket. The gnomon was a string between the two faces. When the string was tight, the two faces formed both a vertical and horizontal sundial. The best material was white ivory, inlaid with black lacquer markings. The best gnomons were black braided silk, linen or hemp. By making the two sundials have different angles to the string (and thus different projections), a diptych can be self-aligning. When both faces show the same time, the diptych shows the local apparent solar time. Additionally, the hinge will be level, and point north (in the northern hemisphere), and the diptych will be angled so the gnomon is parallel to the Earth's axis of rotation. At solar noon, sunrise and sunset, the latitude adjustment of the diptych can't affect the time of either sundial, but at 9am and 3pm, each degree of latitude error (from holding the sundial at the wrong angle) creates four minutes of difference between the two faces. This means that a diptych can also act as a compass and even measure latitude. Some diptychs included a small scale and a plumb-bob to read the latitude. Some others included a compass rose to measure angles to geographic features. Large (meter-sized) diptychs may have been used for navigation in ancient times. navigation

Early 18th Century portable sundials

This form of sundial was about 8 cm diameter and made of brass. It had a brass lid, not shown, to protect it when travelling. Several features enabled precision to be achieved. It had an iron compass needle so that North could be accurately set. The scale division is to 5 minutes. This dial was made in Dublin in 1742 by Gabriel Stokes a mathematical instrument maker.

Elevation sundial

Astrolabes were used as sundials, as well as for calendrical observations, navigation and astronomy. An even smaller design was the ring. It had a small handle, or was a fob or the decoration of a necklace. When held by its handle, a hole would cast a shadow on the inside of the ring, telling the time by markings on the inside. The user had to know if it was morning or evening. Usually the hole was mounted in a sliding lockable piece of metal, which was adjusted to correct date. In recent times, U.S. Special Forces have taken to engraving a simple sundial on their knife-blade. It works even when a watch fails. knife knife

Precision sundials (heliochronometers)

A precision sundial, called a heliochronometer, corrects apparent solar time to mean solar time or another standard time. Heliochronometers usually indicate the minutes to within 1 minute of Universal Time[http://www.sunlitdesign.com/infosearch/sundialaccuracy.htm].

Equatorial bow sundial

The classic shape for a heliochronometer is an equatorial bow sundial. A bar, slot or stretched wire parallel to the earth's axis forms the style. The face is a semicircle with markings on the inner surface. This pattern, built a couple of meters wide out of temperature-invariant steel invar, was used to keep the trains running on time in France before World War I. One of the simplest sundials that reads clock time is an equatorial bow with a gnomon shaped like two vases[http://www.wsanford.com/~wsanford/exo/sundials/ca/claremont/info.html]. The vase-shape directly shades the hour line in the correct place as the year passes, and the sun changes elevation. The most precise sundials ever made are monumental equatorial bows constructed of masonry, part of the Yantra mandir (Jaipur), in India, built as part of a set of astronomical instruments.

Precision noonmarks

In some older houses, particularly farmhouses, a noon-mark can be found carved into a floor or windowsill. Such marks act as sundials to indicate local noon, and they provided a simple and accurate time reference for households that did not possess accurate clocks. In modern times, some Oriental countries' post offices have set their clocks from a precision noon-mark. These in turn provided the times for the rest of the society. The typical noon-mark sundial was a lens set above an analemmatic plate. The plate has an engraved figure-eight shape. When the edge of the sun's image touches the part of the shape for the current month, it is noon!

Ancient Greek sundials

The ancient Greeks used a type of sundial sometimes referred to as pelekinon (axe-like, apparently because shape of the hour and day lines suggest the ancient double-headed ax pelekus). The gnomon was a rod or pole upright in a horizontal face or half-spherical face. The shadow of the tip of the rod sweeps out hyperbolic curves on a flat face, or circles on a spherical face. The advantage of these dials is that they can be marked to tell the exact time for all times of year.

Analemmatic sundials

Analemmatic sundials correct solar time to mean solar time or another standard time. These usually have hour lines shaped like "figure eights" (analemmas) according to the equation of time. This compensates for the slight eccentricity in the Earth's orbit that causes a 15 minute variation from mean solar time. Very accurate dials of this type fit nicely in a public square, using a ball at the tip of a flagpole as the nodus, with the face painted on or inlaid in the pavement. Correction: The two lines above aren't correct. An analemmatic sundial has nothing to do with correction from solar time to standard time. The description below is correct. Correction added october 2005. A fun, less accurate version of the sundial is to lay out the hour marks on concrete, and then let the user stand in a square marked with the month. The month squares are arranged to correct the sundial for the time of year. The user's head then forms the gnomon of the dial. If the sundial is molded into the concrete, it is almost perfectly immune to vandalism, as well as truly fun and reasonably accurate. The geometrical construction of an analemmatic sundial is simple. First imagine an equatorial sundial floating in the air: a vertical bar directed towards the pole and a ring in the plane perpendicular to the bar. Label the lowest point of the ring "12", and the other hour marks as usual. At a certain time and date, the shadow of a certain point A on the bar (which falls here or there depending on the time of year) falls on a certain point B of the ring (which depends on the hour, and the position in the Earth's orbit). Now draw the point B' in the ground just below B and the point A' just below A. Now if you stand at A' your shadow will point at B', because the sun is somewhere in the plane A B A' B'. In middle latitudes, the ellipse with the hour-marks should be about six meters wide, so the shadow of the head of the beholder will fall near it most of the time. Article of interest: [http://pass.maths.org.uk/issue11/features/sundials/index.html “Analemmatic sundials: How to build one and why they work” by C.J. Budd and C.J. Sangwin]

Reflection sundials

Isaac Newton invented a sundial for a south-facing window. He placed a tiny mirror on the windowsill, and painted the sundial's face in a mirror-image pelekinon on the ceiling and walls. The mirror formed the gnomon by reflecting a spot of light. This provides a large, accurate, perfectly correctable sundial with minimal material, and no wasted space at all. This design could easily be made analemmatic.

Analog calculating sundials

A last, interesting variation accurately keeps clock time, while still resembling a conventional garden sundial. It is a horizontal sundial with a face cut on a cardioid (a sort of heart-shape). A cardioid is the shape that connects the intersections between the solar-time marks of a conventional sundial, and the equal-angles of a true clock-time face. The place where the shadow crosses the cardioid's edge is the place where clock time can be read on the underlying clock-time dial. The sundial is adjusted for daylight saving time by rotating the underlying equal-angle clock-time face. The sun-time face does not move.

Digital sundials

A digital sundial uses light and shadow to 'write' the time in numerals (or even words), rather than marking time with position. One such design uses two parallel masks to screen sunlight into patterns appropriate for the time of day.

Reference

Sundials: Their Theory and Construction, Albert E. Waugh, Dover Publications, Inc., 1973, ISBN 0-486-22947-5.

External links

- [http://www.liverpoolmuseums.org.uk/nof/sun/ Sunbeams and Sundials] Guide to the sun, the seasons and sundials
- [http://members.aon.at/sundials/index_e.htm Sundials in and around Austria (Europe)]
- [http://au23.troja.mff.cuni.cz/~mira/sh/sh.php?lang=1 Sundials in the Czech Republic and Slovakia (Europe)]
- Sundials on the Internet, the leading information site on sundials: http://www.sundials.co.uk
- A virtual sundial: http://www.quns.cam.ac.uk/Queens/virtualdial/VirtDial.html
- [http://www.biol.rug.nl/maes/zonnewijzers/welcome-e.htm Sundials]
- [http://www.scottishsundials.co.uk Scottish Sundials - by Location, Type and Date]
- [http://www.digitalsundial.com/patent.html Digital sundial patent, with description of related designs]. The patent was filed June 1995 [http://community.middlebury.edu/~schar/sundial/patent.html].
- [http://www.treveris.com/sonnenuhren Sundials in the oldest city of Germany: Trier]
- [http://planetary.org/rrgtm/marsdial/ MarsDial] Sundials carried on Mars rovers Spirit and Opportunity
- [http://www.infraroth.de/cgi-bin/slinks.pl Sundial links] - collection of sundial links
- The [http://www.sundialsoc.org.uk/ British Sundial Society] has a [http://www.sundialsoc.org.uk/glossary/chronology/chronology.htm history of sundials] and a [http://www.sundialsoc.org.uk/glossary/alpha.htm glossary of sundial terminology]
- [http://www.mysundial.ca/tsp/tsp.html The Sundial Primer] Learn about many different kinds of sundials and how to make them. You will also find a number of different paper sundial kits that are fun to make. Category:Clocks ja:日時計 simple:Sundial

Year

A year is the time between two recurrences of an event related to the orbit of the Earth around the Sun. By extension, this can be applied to any planet: for example, a "Martian year" is a year on Mars.

Seasonal year

A seasonal year is the time between successive recurrences of a seasonal event such as the flooding of a river, the migration of a species of bird, the flowering of a species of plant, the first frost, or the first scheduled game of a certain sport. All of these events can have wide variations of more than a month from year to year.

Calendar year

A calendar year is the time between two dates with the same name in a calendar. Solar calendars usually aim to predict the seasons, but because the length of individual seasonal years varies significantly, they instead use an astronomical year as a surrogate. For example, the ancient Egyptians used the heliacal rising of Sirius to predict the flooding of the Nile. The Gregorian calendar aims to keep the vernal equinox on or close to March 21; hence it follows the vernal equinox year. The average length of its year is 365.2425 days. No astronomical year has an integer number of days or months, so any calendar that follows an astronomical year must have a system of intercalation such as leap years. In the formerly used Julian calendar, the average length of a year was 365.25 days. This is still used as a convenient time unit in astronomy, see below.

Astronomical years

Julian year

The Julian year, as used in astronomy and other sciences, is a time unit defined as exactly 365.25 days. This is the normal meaning of the unit "year" (symbol "a" from the Latin annus, annata) used in various scientific contexts. The Julian century of 36525 days and the Julian millennium of 365250 days are used in astronomical calculations. Fundamentally, expressing a time interval in Julian years is a way to precisely specify how many days (not how many "real" years), for long time intervals where stating the number of days would be unwieldy and unintuitive.

Sidereal year

The sidereal year is the time for the Earth to complete one revolution of its orbit, as measured in a fixed frame of reference (such as the fixed stars, Latin sidus). Its duration in SI days of 86,400 SI seconds each is on average: :365.256 363 051 days (365 d 6 h 9 min 9 s) (at the epoch J2000.0 = 2000 January 1 12:00:00 TT).

Tropical year

A tropical year is the time for the Earth to complete one revolution with respect to the framework provided by the intersection of the ecliptic (the plane of the orbit of the Earth) and the plane of the equator (the plane perpendicular to the rotation axis of the Earth). Because of the precession of the equinoxes, this framework moves slowly westward along the ecliptic with respect to the fixed stars (with a period of about 26,000 tropical years); as a consequence, the Earth completes this year before it completes a full orbit as measured in a fixed reference frame. Therefore a tropical year is shorter than the sidereal year. The exact length of a tropical year depends on the chosen starting point: for example the vernal equinox year is the time between successive vernal equinoxes. The mean tropical year (averaged over all ecliptic points) is: :365.242 189 67 days (365 d 5 h 48 min 45 s) (at the epoch J2000.0).

Anomalistic year

The anomalistic year is the time for the Earth to complete one revolution with respect to its apsides. The orbit of the Earth is elliptical; the extreme points, called apsides, are the perihelion, where the Earth is closest to the Sun (January 2 in 2000), and the aphelion, where the Earth is farthest from the Sun (July 2 in 2000). Because of gravitational disturbances by the other planets, the shape and orientation of the orbit are not fixed, and the apsides slowly move with respect to a fixed frame of reference. Therefore the anomalistic year is slightly longer than the sidereal year. It takes about 112,000 years for the ellipse to revolve once relative to the fixed stars. The anomalistic year is also longer than the tropical year (which calendars attempt to track) and so the date of the perihelion gradually advances every year. It takes about 21,000 years for the ellipse to revolve once relative to the vernal equinox, thus for the date of perihelion to return to the same place (given a calendar that tracks the seasons perfectly). The average duration of the anomalistic year is: :365.259 635 864 days (365 d 6 h 13 min 52 s) (at the epoch J2000.0).

Draconic year

The draconitic year, eclipse year or ecliptic year is the time for the Sun (as seen from the Earth) to complete one revolution with respect to the same lunar node (a point where the Moon's orbit intersects the ecliptic). This period is associated with eclipses: these occur only when both the Sun and the Moon are near these nodes; so eclipses occur within about a month of every half eclipse year. Hence there are two eclipse seasons every eclipse year. The average duration of the eclipse year is: :346.620 075 883 days (346 d 14 h 52 min 54 s) (at the epoch J2000.0). :This term is sometimes also used to designate the time it takes for a complete revolution of the Moon's ascending node around the ecliptic: 18.612 815 932 years (6798.331 019 days).

Fumocy

The full moon cycle or fumocy is the time for the Sun (as seen from the Earth) to complete one revolution with respect to the perigee of the Moon's orbit. This period is associated with the apparent size of the full moon, and also with the varying duration of the anomalistic month. The duration of one full moon cycle is: :411.784 430 29 days (411 d 18 h 49 min 34 s) (at the epoch J2000.0).

Heliacal year

A heliacal year is the interval between the heliacal risings of a star. It equals the sidereal year only if the star is on the ecliptic. It differs from the sidereal year for stars north or south of the ecliptic because of the significant angle (23.5°) between Earth's celestial equator and the ecliptic.

Sothic year

The Sothic year is the interval between heliacal risings of the star Sirius. Its duration is very close to the mean Julian year of 365.25 days.

Gaussian year

The Gaussian year is the sidereal year for a planet of negligible mass (relative to the Sun) and unperturbed by other planets that is governed by the Gaussian gravitational constant. Such a planet would be slightly closer to the Sun than Earth's mean distance. Its length is: :365.256 898 3 days (365 d 6 h 9 min 56 s).

Besselian year

The Besselian year is a tropical year that starts when the fictitious mean Sun reaches an ecliptic longitude of 280°. This is currently on or close to 1 January. It is named after the 19th century German astronomer and mathematician Friedrich Bessel. An approximate formula to compute the current time in Besselian years from the Julian day is: :B = 2000 + (JD - 2451544.53)/365.242189

Great year

The Great year, Platonic year, or Equinoctial cycle corresponds to a complete revolution of the equinoxes around the ecliptic. Its length is approximately 25,770.639 22 years (9,412,725 d 23 h 22 min).

Variation in the length of the year and the day

The exact length of an astronomical year changes over time. The main sources of this change are: #The precession of the equinoxes changes the position of astronomical events with respect to the apsides of Earth's orbit. An event moving toward perihelion recurs with a decreasing period from year to year; an event moving toward aphelion recurs with an increasing period from year to year. #The gravitational influence of the Moon and planets changes the shape of the Earth's orbit. Tidal drag between the Earth and the Moon and Sun increases the length of the day and of the month. This in turn depends on factors such as continental rebound and sea level rise. It is also suspected that changes in the effective mass of the sun, caused by nuclear fusion, could have a significant impact on the earth year over time.

Summary of various kinds of year

- 353, 354 or 355 days — the lengths of regular years in some lunisolar calendars
- 354.37 days — 12 lunar months; the average length of a year in lunar calendars
- 365 days — a common year in many solar calendars; ~31.53 million seconds
- 365.24219 days — a mean tropical year near the year 2000
- 365.2424 days — a vernal equinox year.
- 365.2425 days — the average length of a year in the Gregorian calendar
- 365.25 days — the average length of a year in the Julian calendar; the light year is based on it; it is 31,557,600 seconds
- 365.2564 days — a sidereal year
- 366 days — a leap year in many solar calendars; 31.62 million seconds
- 383, 384 or 385 days — the lengths of leap years in some lunisolar calendars
- 383.9 days — 13 lunar months; a leap year in some lunisolar calendars An average Gregorian year is 365.2425 days = 52.1775 weeks, 8,765.82 hours = 525,949.2 minutes = 31,556,952 seconds (mean solar, not SI). A common year is 365 days = 8,760 hours = 525,600 minutes = 31,536,000 seconds. A leap year is 366 days = 8,784 hours = 527,040 minutes = 31,622,400 seconds. An easy to remember approximation for the number of seconds in a year is

\begin\pi\end

×10⁷ seconds. The 400-year cycle of the Gregorian calendar has 146097 days and hence exactly 20871 weeks. See also Numerical facts about the Gregorian calendar.

Orbit

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

History

Orbits were first analysed mathematically by Johannes Kepler who formulated his results in his laws of planetary motion. He found that the orbits of the planets in our solar system are elliptical, not circular (or epicyclic), as had previously been believed. Isaac Newton demonstrated that Kepler's laws were derivable from his theory of gravitation and that, in general, the orbits of bodies responding to the force of gravity were conic sections. Newton showed that a pair of bodies follow orbits of dimensions that are in inverse proportion to their masses about their common center of mass. Where one body is much more massive than the other, it is a convenient approximation to take the center of mass as coinciding with the center of the more massive body.

Planetary orbits

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

Understanding orbits

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

Newton's laws of motion

For a system of only two bodies that are only influenced by their mutual gravity, their orbits can be exactly calculated by Newton's laws of motion and gravity. Briefly, the sum of the forces will equal the mass times its acceleration. Gravity is proportional to mass, and falls off proportionally to the square of distance. To calculate, it is convenient to describe the motion in a coordinate system that is centered on the heavier body, and we can say that the lighter body is in orbit around the heavier body. An unmoving body that's far from a large object has more energy than one that's close. This is because it can fall farther. This is called "potential energy" because it is not yet actual. With two bodies, an orbit is a flat curve. The orbit can be open (so the object never returns) or closed (returning), depending on the total kinetic + potential energy of the system. In the case of an open orbit, the speed at any position of the orbit is at least the escape velocity for that position, in the case of a closed orbit, always less. The path of a free-falling (orbiting) body is always a conic section. An open orbit has the shape of a hyperbola (or in the limiting case, a parabola); the bodies approach each other for a while, curve around each other around the time of their closest approach, and then separate again forever. This is often the case with comets that occasionally approach the Sun. A closed orbit has the shape of an ellipse (or in the limiting case, a circle). The point where the orbiting body is closest to Earth is the perigee, called periapsis (less properly, "perifocus" or "pericentron") when the orbit is around a body other than Earth. The point where the satellite is farthest from Earth is called apogee, apoapsis, or sometimes apifocus or apocentron. A line drawn from periapsis to apoapsis is the line-of-apsides. This is the major axis of the ellipse, the line through its longest part. Orbiting bodies in closed orbits repeat their path after a constant period of time. This motion is described by the empirical laws of Kepler, which can be mathematically derived from Newton's laws. These can be formulated as follows: # The orbit of a planet around the Sun is an ellipse, with the Sun in one of the focal points of the ellipse. Therefore the orbit lies in a plane, called the orbital plane. The point on the orbit closest to the attracting body is the periapsis. The point farthest from the attracting body is called the apoapsis. There are also specific terms for orbits around particular bodies; things orbiting the Sun have a perihelion and aphelion, things orbiting the Earth have a perigee and apogee, and things orbiting the Moon have a perilune and apolune (or, synonymously, periselene and aposelene). An orbit around any star, not just the Sun, has a periastron and an apastron # As the planet moves around its orbit during a fixed amount of time, the line from Sun to planet sweeps a constant area of the orbital plane, regardless of which part of its orbit the planet traces during that period of time. This means that the planet moves faster near its perihelion than near its aphelion, because at the smaller distance it needs to trace a greater arc to cover the same area. This law is usually stated as "equal areas in equal time." # For each planet, the ratio of the 3rd power of its semi-major axis to the 2nd power of its period is the same constant value for all planets. Except for special cases like Lagrangian points, no method is known to solve the equations of motion for a system with four or more bodies. The 2-body solutions were published by Newton in Principia in 1687. In 1912, K. F. Sundman developed a converging infinite series that solves the 3-body problem, however it converges too slowly to be of much use. Instead, orbits can be approximated with arbitrarily high accuracy. These approximations take two forms. One form takes the pure elliptic motion as a basis, and adds perturbation terms to account for the gravitational influence of multiple bodies. This is convenient for calculating the positions of astronomical bodies. The equations of motion of the moon, planets and other bodies are known with great accuracy, and are used to generate tables for celestial navigation. The differential equation form is used for scientific or mission-planning purposes. According to Newton's laws, the sum of all the forces will equal the mass times its acceleration (F = ma). Therefore accelerations can be expressed in terms of positions. The perturbation terms are much easier to describe in this form. Predicting subsequent positions and velocities from initial ones corresponds to solving an initial value problem. Numerical methods calculate the positions and velocities of the objects a tiny time in the future, then repeat this. However, tiny arithmetic errors from the limited accuracy of a computer's math accumulate, limiting the accuracy of this approach. Differential simulations with large numbers of objects perform the calculations in a hierarchical pairwise fashion between centers of mass. Using this scheme, galaxies, star clusters and other large objects have been simulated.

Analysis of orbital motion

(see also orbit equation and Kepler's first law) To analyse the motion of a body moving under the influence of a force which is always directed towards a fixed point, it is convenient to use polar coordinates with the origin coinciding with the centre of force. In such coordinates the radial and transverse components of the acceleration are, respectively: :

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

and :

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

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

\frac = hu^2

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

\frac + u = \frac

. In the case of an inverse square force law the right hand side of the equation becomes a constant and the equation is seen to be the harmonic equation (up to a shift of origin of the dependent variable). The equation of the orbit described by the particle is thus: :

r = \frac = \frac

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

Orbital parameters

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

Orbital period

See: orbital period

Orbital decay

If some part of a body's orbit enters an atmosphere, its orbit can decay because of drag. At each periapsis, the object scrapes the air, losing energy. Each time, the orbit grows less eccentric (more circular) because the object loses kinetic energy precisely when that energy is at its maximum. Eventually, the orbit circularises and then the object spirals into the atmosphere. The bounds of an atmosphere vary wildly. During solar maxima, the Earth's atmosphere causes drag up to a hundred kilometres higher than during solar minimums. Some satellites with long conductive tethers can also decay because of electromagnetic drag from the Earth's magnetic field. Basically, the wire cuts the magnetic field, and acts as a generator. The wire moves electrons from the near vacuum on one end to the near-vacuum on the other end. The orbital energy is converted to heat in the wire. Another method of artificially influencing an orbit is through the use of solar sails or magnetic sails. These forms of propulsion require no propellant or energy input, and so can be used indefinitely. See statite for one such proposed use. Orbital decay can also occur due to tidal forces for objects below the synchronous orbit for the body they're orbiting. The gravity of the orbiting object raises tidal bulges in the primary, and since below the synchronous orbit the orbiting object is moving faster than the body's surface the bulges lag a short angle behind it. The gravity of the bulges is slightly off of the primary-satellite axis and thus has a component along the satellite's motion. The near bulge slows the object more than the far bulge speeds it up, and as a result the orbit decays. Conversely, the gravity of the satellite on the bulges applies torque on the primary and speeds up its rotation. Artificial satellites are too small to have an appreciable tidal effect on the planets they orbit, but several moons in the solar system are undergoing orbital decay by this mechanism. Mars' innermost moon Phobos is a prime example, and is expected to either impact Mars' surface or break up into a ring within 50 million years. Finally, orbits can decay via the emission of gravitational waves. This mechanism is extremely weak for most stellar objects, only becoming significant in cases where there is a combination of extreme mass and extreme acceleration, such as with black holes or neutron stars that are orbiting each other closely.

Earth orbits

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

Scaling in gravity

The gravitational constant G is defined to be:
- 6.6742 × 10⁻¹¹ N·m²/kg²
- 6.6742 × 10⁻¹¹ m³/(kg·s²)
- 6.6742 × 10⁻¹¹(kg/m³)^-1s^-2. Thus the constant has dimension density^-1 time^-2. This corresponds to the following properties. Scaling of distances (including sizes of bodies, while keeping the densities the same) gives similar orbits without scaling the time: if for example distances are halved, masses are divided by 8, gravitational forces by 16 and gravitational accelerations by 2. Hence orbital periods remain the same. Similarly, when an object is dropped from a tower, the time it takes to fall to the ground remains the same with a scale model of the tower on a scale model of the earth. When all densities are multiplied by four, orbits are the same, but with orbital velocities doubled. When all densities are multiplied by four, and all sizes are halved, orbits are similar, with the same orbital velocities. These properties are illustrated in the formula :

GT^2 \sigma = 3\pi \left( \frac \right)^3,

for an elliptical orbit with semi-major axis a, of a small body around a spherical body with radius r and average density σ, where T is the orbital period.

Role in the evolution of atomic theory

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

External links

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

Kepler's laws of planetary motion

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

Kepler's first law

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

Kepler's second law

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

Kepler's third law

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

P^2 \propto a^3

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

Accuracy and limitations

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

Connection to Newton's laws and conservation laws

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

Kepler's first law

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

\frac = f(r)\hat

. Recall that in polar coordinates: :

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

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

In component form, we have: :

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

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

Since

\dot r = dr/dt

and

\ddot\theta=/

, the latter equation is equivalent to :

\frac = -2\frac

. When integrated, this yields :

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

, :

\ell = r^2\dot\theta

, for some constant

\ell

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

r = \frac

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

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

The equation of motion in the

\hat

direction becomes: :

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

f(r)=-k/r^2

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

\frac + u = \frac

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

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

Replacing u with r and letting θ₀=0: :

r = \frac

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

Kepler's second law

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

\mathbf

of a point mass with mass

m

and velocity

\mathbf

is : :

\mathbf \equiv m \mathbf \times \mathbf

. where

\mathbf

is the position vector of the particle. Since

\mathbf = \frac

, we have: :

\mathbf = \mathbf \times m\frac

taking the time derivative of both sides: :

\frac = \mathbf \times \mathbf = 0

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

|\mathbf|

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

\mathbf

and

d\mathbf

. :

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

Since

|\mathbf|

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

Kepler's third law

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

F=mv^2/r

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

T^2 = \frac \cdot r^3

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

T^2 = \frac \cdot a^3

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

T^2=a^3

Proving Kepler's third law

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

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

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

V_A=V_B\cdot\frac

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

\frac-\frac=\frac-\frac

\frac-\frac=\frac-\frac

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

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

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

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

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

V_B^2 \cdot 4\epsilon=\frac

V_B=\sqrt.

Now that we have

V_B

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

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

However, the total area of the ellipse is equal to

\pi a \sqrta

. (That's the same as

\pi a b

, because

b=\sqrta

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

T\cdot \frac=\pi a \sqrta

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

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

T^2=\fraca^3.

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

M+m

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

M+m

, to give

T^2=\fraca^3

. Q.E.D.

Solution for the motion as a function of time

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

\mboxcs=a\varepsilon

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

\mboxsxz=\frac ba\mboxspz

- y is a point on the circle such that

\mboxcyz=\mboxsxz=\frac ba\mboxspz

and three angles measured from perihelion:
- true anomaly

T=\angle zsp

, the planet as seen from the sun
-