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Astronomy

Astronomy is a natural science that studies celestial objects and the phenomena that occur in the cosmos. It uses mathematics, physics, and chemistry in order to explain their origin and their overall evolution. Objects of interest include planets, moons, stars, nebulae, galaxies, meteoroids, asteroids, and comets. Relevant phenomena include supernova explosions, gamma ray bursts, quasars, blazars, pulsars, and cosmic microwave background radiation. More generally, astronomy studies everything that originates beyond Earth's atmosphere. Cosmology is a branch of astronomy that studies the universe as a whole.

Astronomy is one of the oldest natural sciences. The early civilizations in recorded history made methodical observations of the night sky. These include the Egyptians, Babylonians, Greeks, Indians, Chinese, Maya, and many ancient indigenous peoples of the Americas. In the past, astronomy included disciplines as diverse as astrometry, celestial navigation, observational astronomy, and the making of calendars.

Professional astronomy is split into observational and theoretical branches. Observational astronomy is focused on acquiring data from observations of astronomical objects. This data is then analyzed using basic principles of physics. Theoretical astronomy is oriented toward the development of computer or analytical models to describe astronomical objects and phenomena. These two fields complement each other. Theoretical astronomy seeks to explain observational results and observations are used to confirm theoretical results.

Astronomy is one of the few sciences in which amateurs play an active role. This is especially true for the discovery and observation of transient events. Amateur astronomers have helped with many important discoveries, such as finding new comets.

Etymology
Astronomy (from the Greek ἀστρονομία from ἄστρον astron, "star" and -νομία -nomia from νόμος nomos, "law" or "culture") means "law of the stars" (or "culture of the stars" depending on the translation). Astronomy should not be confused with astrology, the belief system which claims that human affairs are correlated with the positions of celestial objects. Although the two fields share a common origin, they are now entirely distinct.

Use of terms "astronomy" and "astrophysics"
"Astronomy" and "astrophysics" are synonyms. Based on strict dictionary definitions, "astronomy" refers to "the study of objects and matter outside the Earth's atmosphere and of their physical and chemical properties", while "astrophysics" refers to the branch of astronomy dealing with "the behavior, physical properties, and dynamic processes of celestial objects and phenomena". In some cases, as in the introduction of the introductory textbook The Physical Universe by Frank Shu, "astronomy" may be used to describe the qualitative study of the subject, whereas "astrophysics" is used to describe the physics-oriented version of the subject. However, since most modern astronomical research deals with subjects related to physics, modern astronomy could actually be called astrophysics. Some fields, such as astrometry, are purely astronomy rather than also astrophysics. Various departments in which scientists carry out research on this subject may use "astronomy" and "astrophysics", partly depending on whether the department is historically affiliated with a physics department, and many professional astronomers have physics rather than astronomy degrees. Some titles of the leading scientific journals in this field include The Astronomical Journal, The Astrophysical Journal, and Astronomy & Astrophysics.

Ancient times
In early historic times, astronomy only consisted of the observation and predictions of the motions of objects visible to the naked eye. In some locations, early cultures assembled massive artifacts that may have had some astronomical purpose. In addition to their ceremonial uses, these observatories could be employed to determine the seasons, an important factor in knowing when to plant crops and in understanding the length of the year.

Before tools such as the telescope were invented, early study of the stars was conducted using the naked eye. As civilizations developed, most notably in Egypt, Mesopotamia, Greece, Persia, India, China, and Central America, astronomical observatories were assembled and ideas on the nature of the Universe began to develop. Most early astronomy consisted of mapping the positions of the stars and planets, a science now referred to as astrometry. From these observations, early ideas about the motions of the planets were formed, and the nature of the Sun, Moon and the Earth in the Universe were explored philosophically. The Earth was believed to be the center of the Universe with the Sun, the Moon and the stars rotating around it. This is known as the geocentric model of the Universe, or the Ptolemaic system, named after Ptolemy. A particularly important early development was the beginning of mathematical and scientific astronomy, which began among the Babylonians, who laid the foundations for the later astronomical traditions that developed in many other civilizations. The Babylonians discovered that lunar eclipses recurred in a repeating cycle known as a saros.

Following the Babylonians, significant advances in astronomy were made in ancient Greece and the Hellenistic world. Greek astronomy is characterized from the start by seeking a rational, physical explanation for celestial phenomena. In the 3rd century BC, Aristarchus of Samos estimated the size and distance of the Moon and Sun, and he proposed a model of the Solar System where the Earth and planets rotated around the Sun, now called the heliocentric model. In the 2nd century BC, Hipparchus discovered precession, calculated the size and distance of the Moon and invented the earliest known astronomical devices such as the astrolabe. Hipparchus also created a comprehensive catalog of 1020 stars, and most of the constellations of the northern hemisphere derive from Greek astronomy. The Antikythera mechanism (c. 150–80 BC) was an early analog computer designed to calculate the location of the Sun, Moon, and planets for a given date. Technological artifacts of similar complexity did not reappear until the 14th century, when mechanical astronomical clocks appeared in Europe.

Middle Ages
Medieval Europe housed a number of important astronomers. Richard of Wallingford (1292–1336) made major contributions to astronomy and horology, including the invention of the first astronomical clock, the Rectangulus which allowed for the measurement of angles between planets and other astronomical bodies, as well as an equatorium called the Albion which could be used for astronomical calculations such as lunar, solar and planetary longitudes and could predict eclipses. Nicole Oresme (1320–1382) and Jean Buridan (1300–1361) first discussed evidence for the rotation of the Earth, furthermore, Buridan also developed the theory of impetus (predecessor of the modern scientific theory of inertia) which was able to show planets were capable of motion without the intervention of angels. Georg von Peuerbach (1423–1461) and Regiomontanus (1436–1476) helped make astronomical progress instrumental to Copernicus's development of the heliocentric model decades later.

Astronomy flourished in the Islamic world and other parts of the world. This led to the emergence of the first astronomical observatories in the Muslim world by the early 9th century. In 964, the Andromeda Galaxy, the largest galaxy in the Local Group, was described by the Persian Muslim astronomer Abd al-Rahman al-Sufi in his Book of Fixed Stars. The SN 1006 supernova, the brightest apparent magnitude stellar event in recorded history, was observed by the Egyptian Arabic astronomer Ali ibn Ridwan and Chinese astronomers in 1006. Iranian scholar Al-Biruni observed that, contrary to Ptolemy, the Sun's apogee (highest point in the heavens) was mobile, not fixed. Some of the prominent Islamic (mostly Persian and Arab) astronomers who made significant contributions to the science include Al-Battani, Thebit, Abd al-Rahman al-Sufi, Biruni, Abū Ishāq Ibrāhīm al-Zarqālī, Al-Birjandi, and the astronomers of the Maragheh and Samarkand observatories. Astronomers during that time introduced many Arabic names now used for individual stars.

It is also believed that the ruins at Great Zimbabwe and Timbuktu may have housed astronomical observatories. In Post-classical West Africa, Astronomers studied the movement of stars and relation to seasons, crafting charts of the heavens as well as precise diagrams of orbits of the other planets based on complex mathematical calculations. Songhai historian Mahmud Kati documented a meteor shower in August 1583. Europeans had previously believed that there had been no astronomical observation in sub-Saharan Africa during the pre-colonial Middle Ages, but modern discoveries show otherwise.

For over six centuries (from the recovery of ancient learning during the late Middle Ages into the Enlightenment), the Roman Catholic Church gave more financial and social support to the study of astronomy than probably all other institutions. Among the Church's motives was finding the date for Easter.

Scientific revolution
During the Renaissance, Nicolaus Copernicus proposed a heliocentric model of the solar system. His work was defended by Galileo Galilei and expanded upon by Johannes Kepler. Kepler was the first to devise a system that correctly described the details of the motion of the planets around the Sun. However, Kepler did not succeed in formulating a theory behind the laws he wrote down. It was Isaac Newton, with his invention of celestial dynamics and his law of gravitation, who finally explained the motions of the planets. Newton also developed the reflecting telescope.

Improvements in the size and quality of the telescope led to further discoveries. The English astronomer John Flamsteed catalogued over 3000 stars, More extensive star catalogues were produced by Nicolas Louis de Lacaille. The astronomer William Herschel made a detailed catalog of nebulosity and clusters, and in 1781 discovered the planet Uranus, the first new planet found.

During the 18–19th centuries, the study of the three-body problem by Leonhard Euler, Alexis Claude Clairaut, and Jean le Rond d'Alembert led to more accurate predictions about the motions of the Moon and planets. This work was further refined by Joseph-Louis Lagrange and Pierre Simon Laplace, allowing the masses of the planets and moons to be estimated from their perturbations.

Significant advances in astronomy came about with the introduction of new technology, including the spectroscope and photography. Joseph von Fraunhofer discovered about 600 bands in the spectrum of the Sun in 1814–15, which, in 1859, Gustav Kirchhoff ascribed to the presence of different elements. Stars were proven to be similar to the Earth's own Sun, but with a wide range of temperatures, masses, and sizes.

The existence of the Earth's galaxy, the Milky Way, as its own group of stars was only proved in the 20th century, along with the existence of "external" galaxies. The observed recession of those galaxies led to the discovery of the expansion of the Universe. Theoretical astronomy led to speculations on the existence of objects such as black holes and neutron stars, which have been used to explain such observed phenomena as quasars, pulsars, blazars, and radio galaxies. Physical cosmology made huge advances during the 20th century. In the early 1900s the model of the Big Bang theory was formulated, heavily evidenced by cosmic microwave background radiation, Hubble's law, and the cosmological abundances of elements. Space telescopes have enabled measurements in parts of the electromagnetic spectrum normally blocked or blurred by the atmosphere. In February 2016, it was revealed that the LIGO project had detected evidence of gravitational waves in the previous September.

Observational astronomy
The main source of information about celestial bodies and other objects is visible light, or more generally electromagnetic radiation. Observational astronomy may be categorized according to the corresponding region of the electromagnetic spectrum on which the observations are made. Some parts of the spectrum can be observed from the Earth's surface, while other parts are only observable from either high altitudes or outside the Earth's atmosphere. Specific information on these subfields is given below.

Radio astronomy

Radio astronomy uses radiation with wavelengths greater than approximately one millimeter, outside the visible range. Radio astronomy is different from most other forms of observational astronomy in that the observed radio waves can be treated as waves rather than as discrete photons. Hence, it is relatively easier to measure both the amplitude and phase of radio waves, whereas this is not as easily done at shorter wavelengths.

Although some radio waves are emitted directly by astronomical objects, a product of thermal emission, most of the radio emission that is observed is the result of synchrotron radiation, which is produced when electrons orbit magnetic fields. Additionally, a number of spectral lines produced by interstellar gas, notably the hydrogen spectral line at 21 cm, are observable at radio wavelengths.

A wide variety of other objects are observable at radio wavelengths, including supernovae, interstellar gas, pulsars, and active galactic nuclei.

Infrared astronomy

Infrared astronomy is founded on the detection and analysis of infrared radiation, wavelengths longer than red light and outside the range of our vision. The infrared spectrum is useful for studying objects that are too cold to radiate visible light, such as planets, circumstellar disks or nebulae whose light is blocked by dust. The longer wavelengths of infrared can penetrate clouds of dust that block visible light, allowing the observation of young stars embedded in molecular clouds and the cores of galaxies. Observations from the Wide-field Infrared Survey Explorer (WISE) have been particularly effective at unveiling numerous galactic protostars and their host star clusters. With the exception of infrared wavelengths close to visible light, such radiation is heavily absorbed by the atmosphere, or masked, as the atmosphere itself produces significant infrared emission. Consequently, infrared observatories have to be located in high, dry places on Earth or in space. Some molecules radiate strongly in the infrared. This allows the study of the chemistry of space; more specifically it can detect water in comets.

Optical astronomy

Historically, optical astronomy, also called visible light astronomy, is the oldest form of astronomy. Images of observations were originally drawn by hand. In the late 19th century and most of the 20th century, images were made using photographic equipment. Modern images are made using digital detectors, particularly using charge-coupled devices (CCDs) and recorded on modern medium. Although visible light itself extends from approximately 4000 Å to 7000 Å (400 nm to 700 nm), that same equipment can be used to observe some near-ultraviolet and near-infrared radiation.

Ultraviolet astronomy
Ultraviolet astronomy employs ultraviolet wavelengths between approximately 100 and 3200 Å (10 to 320 nm). Light at those wavelengths is absorbed by the Earth's atmosphere, requiring observations at these wavelengths to be performed from the upper atmosphere or from space. Ultraviolet astronomy is best suited to the study of thermal radiation and spectral emission lines from hot blue stars (OB stars) that are very bright in this wave band. This includes the blue stars in other galaxies, which have been the targets of several ultraviolet surveys. Other objects commonly observed in ultraviolet light include planetary nebulae, supernova remnants, and active galactic nuclei. However, as ultraviolet light is easily absorbed by interstellar dust, an adjustment of ultraviolet measurements is necessary.

X-ray astronomy
X-ray astronomy uses X-ray wavelengths. Typically, X-ray radiation is produced by synchrotron emission (the result of electrons orbiting magnetic field lines), thermal emission from thin gases above 107 (10 million) kelvins, and thermal emission from thick gases above 107 Kelvin. Since X-rays are absorbed by the Earth's atmosphere, all X-ray observations must be performed from high-altitude balloons, rockets, or X-ray astronomy satellites. Notable X-ray sources include X-ray binaries, pulsars, supernova remnants, elliptical galaxies, clusters of galaxies, and active galactic nuclei.

Gamma-ray astronomy
Gamma ray astronomy observes astronomical objects at the shortest wavelengths of the electromagnetic spectrum. Gamma rays may be observed directly by satellites such as the Compton Gamma Ray Observatory or by specialized telescopes called atmospheric Cherenkov telescopes. The Cherenkov telescopes do not detect the gamma rays directly but instead detect the flashes of visible light produced when gamma rays are absorbed by the Earth's atmosphere.

Most gamma-ray emitting sources are actually gamma-ray bursts, objects which only produce gamma radiation for a few milliseconds to thousands of seconds before fading away. Only 10% of gamma-ray sources are non-transient sources. These steady gamma-ray emitters include pulsars, neutron stars, and black hole candidates such as active galactic nuclei.

Fields not based on the electromagnetic spectrum
In addition to electromagnetic radiation, a few other events originating from great distances may be observed from the Earth.

In neutrino astronomy, astronomers use heavily shielded underground facilities such as SAGE, GALLEX, and Kamioka II/III for the detection of neutrinos. The vast majority of the neutrinos streaming through the Earth originate from the Sun, but 24 neutrinos were also detected from supernova 1987A. Cosmic rays, which consist of very high energy particles (atomic nuclei) that can decay or be absorbed when they enter the Earth's atmosphere, result in a cascade of secondary particles which can be detected by current observatories. Some future neutrino detectors may also be sensitive to the particles produced when cosmic rays hit the Earth's atmosphere.

Gravitational-wave astronomy is an emerging field of astronomy that employs gravitational-wave detectors to collect observational data about distant massive objects. A few observatories have been constructed, such as the Laser Interferometer Gravitational Observatory LIGO. LIGO made its first detection on 14 September 2015, observing gravitational waves from a binary black hole. A second gravitational wave was detected on 26 December 2015 and additional observations should continue but gravitational waves require extremely sensitive instruments.

The combination of observations made using electromagnetic radiation, neutrinos or gravitational waves and other complementary information, is known as multi-messenger astronomy.

Astrometry and celestial mechanics
One of the oldest fields in astronomy, and in all of science, is the measurement of the positions of celestial objects. Historically, accurate knowledge of the positions of the Sun, Moon, planets and stars has been essential in celestial navigation (the use of celestial objects to guide navigation) and in the making of calendars.

Careful measurement of the positions of the planets has led to a solid understanding of gravitational perturbations, and an ability to determine past and future positions of the planets with great accuracy, a field known as celestial mechanics. More recently the tracking of near-Earth objects will allow for predictions of close encounters or potential collisions of the Earth with those objects.

The measurement of stellar parallax of nearby stars provides a fundamental baseline in the cosmic distance ladder that is used to measure the scale of the Universe. Parallax measurements of nearby stars provide an absolute baseline for the properties of more distant stars, as their properties can be compared. Measurements of the radial velocity and proper motion of stars allow astronomers to plot the movement of these systems through the Milky Way galaxy. Astrometric results are the basis used to calculate the distribution of speculated dark matter in the galaxy.

During the 1990s, the measurement of the stellar wobble of nearby stars was used to detect large extrasolar planets orbiting those stars.

Theoretical astronomy
Theoretical astronomers use several tools including analytical models and computational numerical simulations; each has its particular advantages. Analytical models of a process are better for giving broader insight into the heart of what is going on. Numerical models reveal the existence of phenomena and effects otherwise unobserved.

Theorists in astronomy endeavor to create theoretical models that are based on existing observations and known physics, and to predict observational consequences of those models. The observation of phenomena predicted by a model allows astronomers to select between several alternative or conflicting models. Theorists also modify existing models to take into account new observations. In some cases, a large amount of observational data that is inconsistent with a model may lead to abandoning it largely or completely, as for geocentric theory, the existence of luminiferous aether, and the steady-state model of cosmic evolution.

Phenomena modeled by theoretical astronomers include:
 * stellar dynamics and evolution
 * galaxy formation
 * large-scale distribution of matter in the Universe
 * the origin of cosmic rays
 * general relativity and physical cosmology, including string cosmology and astroparticle physics.

Modern theoretical astronomy reflects dramatic advances in observation since the 1990s, including studies of the cosmic microwave background, distant supernovae and galaxy redshifts, which have led to the development of a standard model of cosmology. This model requires the universe to contain large amounts of dark matter and dark energy whose nature is currently not well understood, but the model gives detailed predictions that are in excellent agreement with many diverse observations.

Astrophysics

Astrophysics is the branch of astronomy that employs the principles of physics and chemistry "to ascertain the nature of the astronomical objects, rather than their positions or motions in space". Among the objects studied are the Sun, other stars, galaxies, extrasolar planets, the interstellar medium and the cosmic microwave background. Their emissions are examined across all parts of the electromagnetic spectrum, and the properties examined include luminosity, density, temperature, and chemical composition. Because astrophysics is a very broad subject, astrophysicists typically apply many disciplines of physics, including mechanics, electromagnetism, statistical mechanics, thermodynamics, quantum mechanics, relativity, nuclear and particle physics, and atomic and molecular physics.

In practice, modern astronomical research often involves a substantial amount of work in the realms of theoretical and observational physics. Some areas of study for astrophysicists include their attempts to determine the properties of dark matter, dark energy, and black holes; whether or not time travel is possible, wormholes can form, or the multiverse exists; and the origin and ultimate fate of the universe. Topics also studied by theoretical astrophysicists include Solar System formation and evolution; stellar dynamics and evolution; galaxy formation and evolution; magnetohydrodynamics; large-scale structure of matter in the universe; origin of cosmic rays; general relativity and physical cosmology, including string cosmology and astroparticle physics.

Astrochemistry
Astrochemistry is the study of the abundance and reactions of molecules in the Universe, and their interaction with radiation. The discipline is an overlap of astronomy and chemistry. The word "astrochemistry" may be applied to both the Solar System and the interstellar medium. The study of the abundance of elements and isotope ratios in Solar System objects, such as meteorites, is also called cosmochemistry, while the study of interstellar atoms and molecules and their interaction with radiation is sometimes called molecular astrophysics. The formation, atomic and chemical composition, evolution and fate of molecular gas clouds is of special interest, because it is from these clouds that solar systems form. Studies in this field contribute to the understanding of the formation of the Solar System, Earth's origin and geology, abiogenesis, and the origin of climate and oceans.

Astrobiology
Astrobiology is an interdisciplinary scientific field concerned with the origins, early evolution, distribution, and future of life in the universe. Astrobiology considers the question of whether extraterrestrial life exists, and how humans can detect it if it does. The term exobiology is similar.

Astrobiology makes use of molecular biology, biophysics, biochemistry, chemistry, astronomy, physical cosmology, exoplanetology and geology to investigate the possibility of life on other worlds and help recognize biospheres that might be different from that on Earth. The origin and early evolution of life is an inseparable part of the discipline of astrobiology. Astrobiology concerns itself with interpretation of existing scientific data, and although speculation is entertained to give context, astrobiology concerns itself primarily with hypotheses that fit firmly into existing scientific theories.

This interdisciplinary field encompasses research on the origin of planetary systems, origins of organic compounds in space, rock-water-carbon interactions, abiogenesis on Earth, planetary habitability, research on biosignatures for life detection, and studies on the potential for life to adapt to challenges on Earth and in outer space.

Physical cosmology
Cosmology (from the Greek κόσμος "world, universe" and λόγος  "word, study" or literally "logic") could be considered the study of the Universe as a whole.

Observations of the large-scale structure of the Universe, a branch known as physical cosmology, have provided a deep understanding of the formation and evolution of the cosmos. Fundamental to modern cosmology is the well-accepted theory of the Big Bang, wherein our Universe began at a single point in time, and thereafter expanded over the course of 13.8 billion years to its present condition. The concept of the Big Bang can be traced back to the discovery of the microwave background radiation in 1965.

In the course of this expansion, the Universe underwent several evolutionary stages. In the very early moments, it is theorized that the Universe experienced a very rapid cosmic inflation, which homogenized the starting conditions. Thereafter, nucleosynthesis produced the elemental abundance of the early Universe. (See also nucleocosmochronology.)

When the first neutral atoms formed from a sea of primordial ions, space became transparent to radiation, releasing the energy viewed today as the microwave background radiation. The expanding Universe then underwent a Dark Age due to the lack of stellar energy sources.

A hierarchical structure of matter began to form from minute variations in the mass density of space. Matter accumulated in the densest regions, forming clouds of gas and the earliest stars, the Population III stars. These massive stars triggered the reionization process and are believed to have created many of the heavy elements in the early Universe, which, through nuclear decay, create lighter elements, allowing the cycle of nucleosynthesis to continue longer.

Gravitational aggregations clustered into filaments, leaving voids in the gaps. Gradually, organizations of gas and dust merged to form the first primitive galaxies. Over time, these pulled in more matter, and were often organized into groups and clusters of galaxies, then into larger-scale superclusters.

Fundamental to the structure of the Universe is the existence of dark matter and dark energy. These are now thought to be its dominant components, forming 96% of the mass of the Universe. For this reason, much effort is expended in trying to understand the physics of these components.

Extragalactic astronomy

The study of objects outside our galaxy is a branch of astronomy concerned with the formation and evolution of galaxies, their morphology (description) and classification, the observation of active galaxies, and at a larger scale, the groups and clusters of galaxies. Finally, the latter is important for the understanding of the large-scale structure of the cosmos.

Most galaxies are organized into distinct shapes that allow for classification schemes. They are commonly divided into spiral, elliptical and Irregular galaxies.

As the name suggests, an elliptical galaxy has the cross-sectional shape of an ellipse. The stars move along random orbits with no preferred direction. These galaxies contain little or no interstellar dust, few star-forming regions, and older stars. Elliptical galaxies may have been formed by other galaxies merging.

A spiral galaxy is organized into a flat, rotating disk, usually with a prominent bulge or bar at the center, and trailing bright arms that spiral outward. The arms are dusty regions of star formation within which massive young stars produce a blue tint. Spiral galaxies are typically surrounded by a halo of older stars. Both the Milky Way and one of our nearest galaxy neighbors, the Andromeda Galaxy, are spiral galaxies.

Irregular galaxies are chaotic in appearance, and are neither spiral nor elliptical. About a quarter of all galaxies are irregular, and the peculiar shapes of such galaxies may be the result of gravitational interaction.

An active galaxy is a formation that emits a significant amount of its energy from a source other than its stars, dust and gas. It is powered by a compact region at the core, thought to be a supermassive black hole that is emitting radiation from in-falling material. A radio galaxy is an active galaxy that is very luminous in the radio portion of the spectrum, and is emitting immense plumes or lobes of gas. Active galaxies that emit shorter frequency, high-energy radiation include Seyfert galaxies, quasars, and blazars. Quasars are believed to be the most consistently luminous objects in the known universe.

The large-scale structure of the cosmos is represented by groups and clusters of galaxies. This structure is organized into a hierarchy of groupings, with the largest being the superclusters. The collective matter is formed into filaments and walls, leaving large voids between.

Galactic astronomy

The Solar System orbits within the Milky Way, a barred spiral galaxy that is a prominent member of the Local Group of galaxies. It is a rotating mass of gas, dust, stars and other objects, held together by mutual gravitational attraction. As the Earth is located within the dusty outer arms, there are large portions of the Milky Way that are obscured from view.

In the center of the Milky Way is the core, a bar-shaped bulge with what is believed to be a supermassive black hole at its center. This is surrounded by four primary arms that spiral from the core. This is a region of active star formation that contains many younger, population I stars. The disk is surrounded by a spheroid halo of older, population II stars, as well as relatively dense concentrations of stars known as globular clusters.

Between the stars lies the interstellar medium, a region of sparse matter. In the densest regions, molecular clouds of molecular hydrogen and other elements create star-forming regions. These begin as a compact pre-stellar core or dark nebulae, which concentrate and collapse (in volumes determined by the Jeans length) to form compact protostars.

As the more massive stars appear, they transform the cloud into an H II region (ionized atomic hydrogen) of glowing gas and plasma. The stellar wind and supernova explosions from these stars eventually cause the cloud to disperse, often leaving behind one or more young open clusters of stars. These clusters gradually disperse, and the stars join the population of the Milky Way.

Kinematic studies of matter in the Milky Way and other galaxies have demonstrated that there is more mass than can be accounted for by visible matter. A dark matter halo appears to dominate the mass, although the nature of this dark matter remains undetermined.

Stellar astronomy

The study of stars and stellar evolution is fundamental to our understanding of the Universe. The astrophysics of stars has been determined through observation and theoretical understanding; and from computer simulations of the interior. Star formation occurs in dense regions of dust and gas, known as giant molecular clouds. When destabilized, cloud fragments can collapse under the influence of gravity, to form a protostar. A sufficiently dense, and hot, core region will trigger nuclear fusion, thus creating a main-sequence star.

Almost all elements heavier than hydrogen and helium were created inside the cores of stars.

The characteristics of the resulting star depend primarily upon its starting mass. The more massive the star, the greater its luminosity, and the more rapidly it fuses its hydrogen fuel into helium in its core. Over time, this hydrogen fuel is completely converted into helium, and the star begins to evolve. The fusion of helium requires a higher core temperature. A star with a high enough core temperature will push its outer layers outward while increasing its core density. The resulting red giant formed by the expanding outer layers enjoys a brief life span, before the helium fuel in the core is in turn consumed. Very massive stars can also undergo a series of evolutionary phases, as they fuse increasingly heavier elements.

The final fate of the star depends on its mass, with stars of mass greater than about eight times the Sun becoming core collapse supernovae; while smaller stars blow off their outer layers and leave behind the inert core in the form of a white dwarf. The ejection of the outer layers forms a planetary nebula. The remnant of a supernova is a dense neutron star, or, if the stellar mass was at least three times that of the Sun, a black hole. Closely orbiting binary stars can follow more complex evolutionary paths, such as mass transfer onto a white dwarf companion that can potentially cause a supernova. Planetary nebulae and supernovae distribute the "metals" produced in the star by fusion to the interstellar medium; without them, all new stars (and their planetary systems) would be formed from hydrogen and helium alone.

Solar astronomy

At a distance of about eight light-minutes, the most frequently studied star is the Sun, a typical main-sequence dwarf star of stellar class G2 V, and about 4.6 billion years (Gyr) old. The Sun is not considered a variable star, but it does undergo periodic changes in activity known as the sunspot cycle. This is an 11-year oscillation in sunspot number. Sunspots are regions of lower-than- average temperatures that are associated with intense magnetic activity.

The Sun has steadily increased in luminosity by 40% since it first became a main-sequence star. The Sun has also undergone periodic changes in luminosity that can have a significant impact on the Earth. The Maunder minimum, for example, is believed to have caused the Little Ice Age phenomenon during the Middle Ages.

At the center of the Sun is the core region, a volume of sufficient temperature and pressure for nuclear fusion to occur. Above the core is the radiation zone, where the plasma conveys the energy flux by means of radiation. Above that is the convection zone where the gas material transports energy primarily through physical displacement of the gas known as convection. It is believed that the movement of mass within the convection zone creates the magnetic activity that generates sunspots. The visible outer surface of the Sun is called the photosphere. Above this layer is a thin region known as the chromosphere. This is surrounded by a transition region of rapidly increasing temperatures, and finally by the super-heated corona.

A solar wind of plasma particles constantly streams outward from the Sun until, at the outermost limit of the Solar System, it reaches the heliopause. As the solar wind passes the Earth, it interacts with the Earth's magnetic field (magnetosphere) and deflects the solar wind, but traps some creating the Van Allen radiation belts that envelop the Earth. The aurora are created when solar wind particles are guided by the magnetic flux lines into the Earth's polar regions where the lines then descend into the atmosphere.

Planetary science

Planetary science is the study of the assemblage of planets, moons, dwarf planets, comets, asteroids, and other bodies orbiting the Sun, as well as extrasolar planets. The Solar System has been relatively well-studied, initially through telescopes and then later by spacecraft. This has provided a good overall understanding of the formation and evolution of the Sun's planetary system, although many new discoveries are still being made.

The Solar System is divided into the inner Solar System (subdivided into the inner planets and the asteroid belt), the outer Solar System (subdivided into the outer planets and centaurs), comets, the trans-Neptunian region (subdivided into the Kuiper belt, and the scattered disc) and the farthest regions (e.g., boundaries of the heliosphere, and the Oort Cloud, which may extend as far as a light-year). The inner terrestrial planets consist of Mercury, Venus, Earth, and Mars. The outer giant planets are the gas giants (Jupiter and Saturn) and the ice giants (Uranus and Neptune).

The planets were formed 4.6 billion years ago in the protoplanetary disk that surrounded the early Sun. Through a process that included gravitational attraction, collision, and accretion, the disk formed clumps of matter that, with time, became protoplanets. The radiation pressure of the solar wind then expelled most of the unaccreted matter, and only those planets with sufficient mass retained their gaseous atmosphere. The planets continued to sweep up, or eject, the remaining matter during a period of intense bombardment, evidenced by the many impact craters on the Moon. During this period, some of the protoplanets may have collided and one such collision may have formed the Moon.

Once a planet reaches sufficient mass, the materials of different densities segregate within, during planetary differentiation. This process can form a stony or metallic core, surrounded by a mantle and an outer crust. The core may include solid and liquid regions, and some planetary cores generate their own magnetic field, which can protect their atmospheres from solar wind stripping.

A planet or moon's interior heat is produced from the collisions that created the body, by the decay of radioactive materials (e.g. uranium, thorium, and 26Al), or tidal heating caused by interactions with other bodies. Some planets and moons accumulate enough heat to drive geologic processes such as volcanism and tectonics. Those that accumulate or retain an atmosphere can also undergo surface erosion from wind or water. Smaller bodies, without tidal heating, cool more quickly; and their geological activity ceases with the exception of impact cratering.

Interdisciplinary studies
Astronomy and astrophysics have developed significant interdisciplinary links with other major scientific fields. Archaeoastronomy is the study of ancient or traditional astronomies in their cultural context, utilizing archaeological and anthropological evidence. Astrobiology is the study of the advent and evolution of biological systems in the Universe, with particular emphasis on the possibility of non-terrestrial life. Astrostatistics is the application of statistics to astrophysics to the analysis of a vast amount of observational astrophysical data.

The study of chemicals found in space, including their formation, interaction and destruction, is called astrochemistry. These substances are usually found in molecular clouds, although they may also appear in low-temperature stars, brown dwarfs and planets. Cosmochemistry is the study of the chemicals found within the Solar System, including the origins of the elements and variations in the isotope ratios. Both of these fields represent an overlap of the disciplines of astronomy and chemistry. As "forensic astronomy", finally, methods from astronomy have been used to solve problems of art history and occasionally of law.

Amateur astronomy

Astronomy is one of the sciences to which amateurs can contribute the most.

Collectively, amateur astronomers observe a variety of celestial objects and phenomena sometimes with consumer-level equipment or equipment that they build themselves. Common targets of amateur astronomers include the Sun, the Moon, planets, stars, comets, meteor showers, and a variety of deep-sky objects such as star clusters, galaxies, and nebulae. Astronomy clubs are located throughout the world and many have programs to help their members set up and complete observational programs including those to observe all the objects in the Messier (110 objects) or Herschel 400 catalogues of points of interest in the night sky. One branch of amateur astronomy, astrophotography, involves the taking of photos of the night sky. Many amateurs like to specialize in the observation of particular objects, types of objects, or types of events that interest them.

Most amateurs work at visible wavelengths, but many experiment with wavelengths outside the visible spectrum. This includes the use of infrared filters on conventional telescopes, and also the use of radio telescopes. The pioneer of amateur radio astronomy was Karl Jansky, who started observing the sky at radio wavelengths in the 1930s. A number of amateur astronomers use either homemade telescopes or use radio telescopes which were originally built for astronomy research but which are now available to amateurs (e.g. the One-Mile Telescope).

Amateur astronomers continue to make scientific contributions to the field of astronomy and it is one of the few scientific disciplines where amateurs can still make significant contributions. Amateurs can make occultation measurements that are used to refine the orbits of minor planets. They can also discover comets, and perform regular observations of variable stars. Improvements in digital technology have allowed amateurs to make impressive advances in the field of astrophotography.

Unsolved problems in astronomy
In the 21st century there remain important unanswered questions in astronomy. Some are cosmic in scope: for example, what are dark matter and dark energy? These dominate the evolution and fate of the cosmos, yet their true nature remains unknown. What will be the ultimate fate of the universe? Why is the abundance of lithium in the cosmos four times lower than predicted by the standard Big Bang model? Others pertain to more specific classes of phenomena. For example, is the Solar System normal or atypical? What is the origin of the stellar mass spectrum? That is, why do astronomers observe the same distribution of stellar masses—the initial mass function—apparently regardless of the initial conditions? Likewise, questions remain about the formation of the first galaxies, the origin of supermassive black holes, the source of ultra-high-energy cosmic rays, and more.

Is there other life in the Universe? Especially, is there other intelligent life? If so, what is the explanation for the Fermi paradox? The existence of life elsewhere has important scientific and philosophical implications.

title: Astronomy


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Talk:Astronomy

Wiki Education Foundation-supported course assignment
This article is or was the subject of a Wiki Education Foundation-supported course assignment. Further details are available on the course page. Student editor(s): Natalie.Lynne.

Above undated message substituted from Template:Dashboard.wikiedu.org assignment by PrimeBOT (talk) 14:54, 16 January 2022 (UTC)

Discussion - Article for deletion on Modern Mars habitability
There is an article at AfD that may interest you. Please comment at WP:Articles for deletion/Modern Mars habitability — Preceding unsigned comment added by Robertinventor (talk • contribs) 13:46, 18 August 2018 (UTC)

Issues with sources and overall eval
There seem to be sources that take me to an organization's website, but not necessarily any information pertaining to the main topic of the page itself. example: http://www.sydneyobservatory.com.au/[permanent dead link] Official Web Site of the Sydney Observatory

It would also be beneficial, I think, to include as much of the earliest instruments that were used in astronomy by early astronomers like Copernicus, Aristotle, etc.

Overall, the article is relatively neutral and does explain the basic idea of astronomy. 107.77.199.213 (talk) 15:08, 31 October 2018 (UTC) — Preceding unsigned comment added by 107.77.199.213 (talk) 15:07, 31 October 2018 (UTC)

Stagnant in medieval Europe? Not really
The article, as it is, offers absolutely zero space to any of the astronomical progress in medieval Europe on the basis that it was "stagnant", in other words, implying there was none. But this is an old view and modern historians are certainly aware of progress in astronomy in Europe during the Middle Ages, if not major. The page fails to name a number of medieval European astronomers who made contributions to the study of astronomy, including William of Hirsau, Georg von Peuerbach, or even Richard of Wallingford who, among many things, invented the first astronomical clock.64.231.43.14 (talk) 23:08, 2 December 2018 (UTC)
 * The statement that you removed: "During the Middle Ages, astronomy was mostly stagnant in medieval Europe, at least until the 13th century" does really contradict what you added - all these astronomers lived in the 13-th century or later. So, I reinstated it. Ruslik_ Zero 20:30, 3 December 2018 (UTC)

Distance to a star
Someone asked if this article was ready for a FAR. That inspired me to look at the article, which I had not looked at before. I'd like to discuss one sentence:

The first obvious question is why would someone include a statement that someone measured the distance to a star. This happens all the time, so mentioning this example would be encyclopedic if it were notable in some way - the first, or especially significant for some reason.

I looked at 61 Cygni and it does state:

That makes it worthy of mention, but that article doesn't have a source. (There are cites at the end of the paragraph, but they support other facts.)

The Astronomy article does have a source: Forbes, 1909, pp. 147–50

However, that source states:

So while the 61 Cygni has a date of 1838, that date is not supported in that article, and the source in the astronomy article gives a range, not a specific year. I won't be surprised if 1838 is the right date, but a proper source is needed.

The 61 Cygni article claims it was the first distance estimate (other than the sun), which justifies the inclusion of the statement, but again, that article is unsourced. The source in the astronomy article says:

That means Bessel wasn't the first, although he was one of the first. Henderson's calculation in 1832–33 predate Bessel's. Perhaps there's a reason to mention Bessel while not mentioning Henderson, but we need some rationale.

The Wikipedia article on Henderson is Thomas Henderson (astronomer). It states that he was:

That article suggests he did the calculations before Bessel, but did not publish until after Bessel, (in 1839) which may be why some credit Bessel as the first.

This source: A letter from Professor Bessel to Sir J. Herschel may help nail down the date, but 1838 is clearly the date of the letter. The source talks about his work in 1834, and 1837, but just when he starts talking about the final steps, the excerpt ends, so we need someone with access to the full article to see whether he completed the task in 1837, or 1838, or whether convention suggests that the discovery date is the date of publication.

It might be that Bessel deserves the credit for first publication, but, even if so, the sentence in this article refers to the "first star to have its stellar parallax measured" which isn't literally true.

I don't doubt that there is a relatively easy fix, but I do suggest that the current statement may be literally wrong, and if not wrong, is not supported by the citations provided, plus needs some motivation for inclusion.-- S Philbrick (Talk)  23:43, 31 May 2020 (UTC)

In 1995 the solr and hemispheric observatory (soho) 24.112.118.245 (talk) 00:15, 9 May 2022 (UTC)

Wiki Education assignment: CMN2160B
— Assignment last updated by Sugar Tian (talk) 01:15, 2 October 2022 (UTC)

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Talk:Astronomy


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Uniform convergence

In the mathematical field of analysis, uniform convergence is a mode of convergence of functions stronger than pointwise convergence. A sequence of functions $$(f_n)$$ converges uniformly to a limiting function $$f$$ on a set $$E$$ as the function domain if, given any arbitrarily small positive number $$\epsilon$$, a number $$N$$ can be found such that each of the functions $$f_N, f_{N+1},f_{N+2},\ldots$$ differs from $$f$$ by no more than $$\epsilon$$ at every point $$x$$ in $$E$$. Described in an informal way, if $$f_n$$ converges to $$f$$ uniformly, then how quickly the functions $$f_n$$ approach $$f$$ is "uniform" throughout $$E$$ in the following sense: in order to guarantee that $$f_n(x)$$ differs from $$f(x)$$ by less than a chosen distance $$\epsilon$$, we only need to make sure that $$n$$ is larger than or equal to a certain $$N$$, which we can find without knowing the value of $$x\in E$$ in advance. In other words, there exists a number $$N=N(\epsilon)$$ that could depend on $$\epsilon$$ but is independent of $$x$$, such that choosing $$n\geq N$$ will ensure that $$|f_n(x)-f(x)|<\epsilon$$ for all $$x\in E$$. In contrast, pointwise convergence of $$f_n$$ to $$f$$ merely guarantees that for any $$x\in E$$ given in advance, we can find $$N=N(\epsilon, x)$$ (i.e., $$N$$ could depend on the values of both $$\epsilon$$ and $$x$$) such that, for that particular $$x$$, $$f_n(x)$$ falls within $$\epsilon$$ of $$f(x)$$ whenever $$n\geq N$$ (and a different $$x$$ may require a different, larger $$N$$ for $$n\geq N$$ to guarantee that $$|f_n(x)-f(x)|<\epsilon$$).

The difference between uniform convergence and pointwise convergence was not fully appreciated early in the history of calculus, leading to instances of faulty reasoning. The concept, which was first formalized by Karl Weierstrass, is important because several properties of the functions $$f_n$$, such as continuity, Riemann integrability, and, with additional hypotheses, differentiability, are transferred to the limit $$f$$ if the convergence is uniform, but not necessarily if the convergence is not uniform.

History
In 1821 Augustin-Louis Cauchy published a proof that a convergent sum of continuous functions is always continuous, to which Niels Henrik Abel in 1826 found purported counterexamples in the context of Fourier series, arguing that Cauchy's proof had to be incorrect. Completely standard notions of convergence did not exist at the time, and Cauchy handled convergence using infinitesimal methods. When put into the modern language, what Cauchy proved is that a uniformly convergent sequence of continuous functions has a continuous limit. The failure of a merely pointwise-convergent limit of continuous functions to converge to a continuous function illustrates the importance of distinguishing between different types of convergence when handling sequences of functions.

The term uniform convergence was probably first used by Christoph Gudermann, in an 1838 paper on elliptic functions, where he employed the phrase "convergence in a uniform way" when the "mode of convergence" of a series $\sum_{n=1}^\infty f_n(x,\phi,\psi)$ is independent of the variables $$\phi$$ and $$\psi.$$ While he thought it a "remarkable fact" when a series converged in this way, he did not give a formal definition, nor use the property in any of his proofs.

Later Gudermann's pupil Karl Weierstrass, who attended his course on elliptic functions in 1839–1840, coined the term gleichmäßig konvergent (uniformly convergent) which he used in his 1841 paper Zur Theorie der Potenzreihen, published in 1894. Independently, similar concepts were articulated by Philipp Ludwig von Seidel and George Gabriel Stokes. G. H. Hardy compares the three definitions in his paper "Sir George Stokes and the concept of uniform convergence" and remarks: "Weierstrass's discovery was the earliest, and he alone fully realized its far-reaching importance as one of the fundamental ideas of analysis."

Under the influence of Weierstrass and Bernhard Riemann this concept and related questions were intensely studied at the end of the 19th century by Hermann Hankel, Paul du Bois-Reymond, Ulisse Dini, Cesare Arzelà and others.

Definition
We first define uniform convergence for real-valued functions, although the concept is readily generalized to functions mapping to metric spaces and, more generally, uniform spaces (see below).

Suppose $$E$$ is a set and $$(f_n)_{n \in \N}$$ is a sequence of real-valued functions on it. We say the sequence $$(f_n)_{n \in \N}$$ is uniformly convergent on $$E$$ with limit $$f: E \to \R$$ if for every $$\epsilon > 0,$$ there exists a natural number $$N$$ such that for all $$n \geq N$$ and for all $$x \in E$$

 * $$|f_n(x)-f(x)|<\epsilon.$$

The notation for uniform convergence of $$f_n$$ to $$f$$ is not quite standardized and different authors have used a variety of symbols, including (in roughly decreasing order of popularity):

 * $$f_n\rightrightarrows f, \quad \underset{n\to\infty}{\mathrm{unif\ lim}}f_n = f, \quad f_n \overset{\mathrm{unif.}}{\longrightarrow} f, \quad f=\mathrm{u}\!\!-\!\!\!\lim_{n\to\infty} f_n .$$

Frequently, no special symbol is used, and authors simply write

 * $$f_n\to f \quad \mathrm{uniformly}$$

to indicate that convergence is uniform. (In contrast, the expression $$f_n\to f$$ on $$E$$ without an adverb is taken to mean pointwise convergence on $$E$$: for all $$ x \in E $$, $$f_n(x)\to f(x)$$ as $$n\to\infty$$.)

Since $$\R$$ is a complete metric space, the Cauchy criterion can be used to give an equivalent alternative formulation for uniform convergence: $$(f_n)_{n\in\N}$$ converges uniformly on $$E$$ (in the previous sense) if and only if for every $$ \epsilon > 0 $$, there exists a natural number $$N$$ such that

 * $$x\in E, m,n\geq N \implies |f_m(x)-f_n(x)|<\epsilon$$.

In yet another equivalent formulation, if we define

 * $$ d_n = \sup_{x\in E} |f_n(x) - f(x) |,$$

then $$ f_n $$ converges to $$f$$ uniformly if and only if $$d_n\to 0$$ as $$n\to\infty$$. Thus, we can characterize uniform convergence of $$(f_n)_{n \in \N}$$ on $$E$$ as (simple) convergence of $$(f_n)_{n \in \N}$$ in the function space $$\R^E$$ with respect to the uniform metric (also called the supremum metric), defined by

 * $$d(f,g)=\sup_{x\in E} |f(x)-g(x)|.$$

Symbolically,

 * $$f_n\rightrightarrows f\iff d(f_n,f) \to 0$$.

The sequence $$(f_n)_{n \in \N}$$ is said to be locally uniformly convergent with limit $$f$$ if $$E $$ is a metric space and for every $$x\in E$$, there exists an $$r > 0$$ such that $$(f_n)$$ converges uniformly on $$B(x,r)\cap E.$$ It is clear that uniform convergence implies local uniform convergence, which implies pointwise convergence.

Generalizations
One may straightforwardly extend the concept to functions E → M, where (M, d) is a metric space, by replacing $$|f_n(x)-f(x)|$$ with $$d(f_n(x),f(x))$$.

The most general setting is the uniform convergence of nets of functions E → X, where X is a uniform space. We say that the net $$(f_\alpha)$$ converges uniformly with limit f : E → X if and only if for every entourage V in X, there exists an $$\alpha_0$$, such that for every x in E and every $$\alpha\geq \alpha_0$$, $$(f_\alpha(x),f(x))$$ is in V. In this situation, uniform limit of continuous functions remains continuous.

Definition in a hyperreal setting
Uniform convergence admits a simplified definition in a hyperreal setting. Thus, a sequence $$f_n$$ converges to f uniformly if for all hyperreal x in the domain of $$f^*$$ and all infinite n, $$f_n^*(x)$$ is infinitely close to $$f^*(x)$$ (see microcontinuity for a similar definition of uniform continuity). In contrast, pointwise continuity requires this only for real x.

Examples
For $$x \in [0,1)$$, a basic example of uniform convergence can be illustrated as follows: the sequence $$(1/2)^{x+n}$$ converges uniformly, while $$x^n$$ does not. Specifically, assume $$\epsilon=1/4$$. Each function $$(1/2)^{x+n}$$ is less than or equal to $$1/4$$ when $$n \geq 2$$, regardless of the value of $$x$$. On the other hand, $$x^n$$ is only less than or equal to $$1/4$$ at ever increasing values of $$n$$ when values of $$x$$ are selected closer and closer to 1 (explained more in depth further below).

Given a topological space X, we can equip the space of bounded real or complex-valued functions over X with the uniform norm topology, with the uniform metric defined by

 * $$d(f,g)=\|f-g\|_{\infty}=\sup_{x\in X} |f(x)-g(x)|.$$

Then uniform convergence simply means convergence in the uniform norm topology:

 * $$\lim_{n\to\infty}\|f_n-f\|_{\infty}=0$$.

The sequence of functions $$(f_n)$$

 * $$\begin{cases} f_n:[0,1]\to [0,1] \\ f_n(x)=x^n \end{cases}$$

is a classic example of a sequence of functions that converges to a function $$f$$ pointwise but not uniformly. To show this, we first observe that the pointwise limit of $$(f_n)$$ as $$n\to\infty$$ is the function $$f$$, given by

 * $$f(x) = \lim_{n\to \infty} f_n(x) = \begin{cases} 0, & x \in [0,1); \\ 1, & x=1. \end{cases} $$

Pointwise convergence: Convergence is trivial for $$x=0$$ and $$x=1$$, since $$f_n(0)=f(0)=0$$ and $$f_n(1)=f(1)=1$$, for all $$n$$. For $$x \in (0,1)$$ and given $$\epsilon>0$$, we can ensure that $$|f_n(x)-f(x)|<\epsilon$$ whenever $$n\geq N$$ by choosing $$N = \lceil\log\epsilon/\log x\rceil$$, which is the minimum integer exponent of $$x$$ that allows it to reach or dip below $$\epsilon$$ (here the upper square brackets indicate rounding up, see ceiling function). Hence, $$f_n\to f$$ pointwise for all $$x\in[0,1]$$. Note that the choice of $$N$$ depends on the value of $$\epsilon$$ and $$x$$. Moreover, for a fixed choice of $$\epsilon$$, $$N$$ (which cannot be defined to be smaller) grows without bound as $$x$$ approaches 1. These observations preclude the possibility of uniform convergence.

Non-uniformity of convergence: The convergence is not uniform, because we can find an $$\epsilon>0$$ so that no matter how large we choose $$N,$$ there will be values of $$x \in [0,1]$$ and $$n \geq N$$ such that $$|f_n(x)-f(x)|\geq\epsilon.$$ To see this, first observe that regardless of how large $$n$$ becomes, there is always an $$x_0 \in [0,1)$$ such that $$f_n(x_0)=1/2.$$ Thus, if we choose $$\epsilon = 1/4,$$ we can never find an $$N$$ such that $$|f_n(x)-f(x)|<\epsilon$$ for all $$x\in[0,1]$$ and $$n\geq N$$. Explicitly, whatever candidate we choose for $$N$$, consider the value of $$f_N$$ at $$x_0 = (1/2)^{1/N}$$. Since

 * $$\left|f_N(x_0) - f(x_0)\right| = \left| \left[ \left(\frac{1}{2}\right)^{\frac{1}{N}} \right]^N - 0 \right| = \frac{1}{2} > \frac{1}{4} = \epsilon,$$

the candidate fails because we have found an example of an $$x\in[0,1]$$ that "escaped" our attempt to "confine" each $$f_n\ (n\geq N)$$ to within $$\epsilon$$ of $$f $$ for all $$x\in[0,1]$$. In fact, it is easy to see that
 * $$\lim_{n\to\infty}\|f_n-f\|_{\infty}=1,$$

contrary to the requirement that $$\|f_n-f\|_{\infty}\to 0$$ if $$f_n \rightrightarrows f$$.

In this example one can easily see that pointwise convergence does not preserve differentiability or continuity. While each function of the sequence is smooth, that is to say that for all n, $$f_n\in C^{\infty}([0,1])$$, the limit $$\lim_{n\to \infty}f_n$$ is not even continuous.

Exponential function
The series expansion of the exponential function can be shown to be uniformly convergent on any bounded subset $$S \subset \C$$ using the Weierstrass M-test.

Theorem (Weierstrass M-test). Let $$(f_n)$$ be a sequence of functions $$f_n:E\to \C$$ and let $$M_n $$ be a sequence of positive real numbers such that $$|f_n(x)|\le M_n$$ for all $$x\in E$$ and $$n=1,2, 3, \ldots$$ If $\sum_n M_n$ converges, then $\sum_n f_n$  converges absolutely and uniformly on $$E$$.

The complex exponential function can be expressed as the series:

 * $$\sum_{n=0}^{\infty}\frac{z^n}{n!}.$$

Any bounded subset is a subset of some disc $$D_R$$ of radius $$R,$$ centered on the origin in the complex plane. The Weierstrass M-test requires us to find an upper bound $$M_n$$ on the terms of the series, with $$M_n$$ independent of the position in the disc:

 * $$\left| \frac{z^n}{n!} \right|\le M_n, \forall z\in D_R.$$

To do this, we notice

 * $$\left| \frac{z^n}{n!}\right| \le \frac{|z|^n}{n!} \le \frac{R^n}{n!}$$

and take $$M_n=\tfrac{R^n}{n!}.$$

If $$\sum_{n=0}^{\infty}M_n$$ is convergent, then the M-test asserts that the original series is uniformly convergent.

The ratio test can be used here:

 * $$\lim_{n \to \infty}\frac{M_{n+1}}{M_n}=\lim_{n \to \infty}\frac{R^{n+1}}{R^n}\frac{n!}{(n+1)!}=\lim_{n \to \infty}\frac{R}{n+1}=0$$

which means the series over $$M_n$$ is convergent. Thus the original series converges uniformly for all $$z\in D_R,$$ and since $$S\subset D_R$$, the series is also uniformly convergent on $$S.$$

Properties

 * Every uniformly convergent sequence is locally uniformly convergent.
 * Every locally uniformly convergent sequence is compactly convergent.
 * For locally compact spaces local uniform convergence and compact convergence coincide.
 * A sequence of continuous functions on metric spaces, with the image metric space being complete, is uniformly convergent if and only if it is uniformly Cauchy.
 * If $$S$$ is a compact interval (or in general a compact topological space), and $$ (f_n)$$ is a monotone increasing sequence (meaning $$ f_n(x) \leq f_{n+1}(x)$$ for all n and x) of continuous functions with a pointwise limit $$ f$$ which is also continuous, then the convergence is necessarily uniform (Dini's theorem). Uniform convergence is also guaranteed if $$ S$$ is a compact interval and $$(f_n)$$ is an equicontinuous sequence that converges pointwise.

To continuity

If $$E$$ and $$M$$ are topological spaces, then it makes sense to talk about the continuity of the functions $$f_n,f:E\to M$$. If we further assume that $$M$$ is a metric space, then (uniform) convergence of the $$f_n$$ to $$f$$ is also well defined. The following result states that continuity is preserved by uniform convergence:

$$

This theorem is proved by the "$ε/3$ trick", and is the archetypal example of this trick: to prove a given inequality ($ε$), one uses the definitions of continuity and uniform convergence to produce 3 inequalities ($ε/3$), and then combines them via the triangle inequality to produce the desired inequality.

This theorem is an important one in the history of real and Fourier analysis, since many 18th century mathematicians had the intuitive understanding that a sequence of continuous functions always converges to a continuous function. The image above shows a counterexample, and many discontinuous functions could, in fact, be written as a Fourier series of continuous functions. The erroneous claim that the pointwise limit of a sequence of continuous functions is continuous (originally stated in terms of convergent series of continuous functions) is infamously known as "Cauchy's wrong theorem". The uniform limit theorem shows that a stronger form of convergence, uniform convergence, is needed to ensure the preservation of continuity in the limit function.

More precisely, this theorem states that the uniform limit of uniformly continuous functions is uniformly continuous; for a locally compact space, continuity is equivalent to local uniform continuity, and thus the uniform limit of continuous functions is continuous.

To differentiability
If $$S$$ is an interval and all the functions $$f_n$$ are differentiable and converge to a limit $$f$$, it is often desirable to determine the derivative function $$f'$$ by taking the limit of the sequence $$f'_n$$. This is however in general not possible: even if the convergence is uniform, the limit function need not be differentiable (not even if the sequence consists of everywhere-analytic functions, see Weierstrass function), and even if it is differentiable, the derivative of the limit function need not be equal to the limit of the derivatives. Consider for instance $$f_n(x) = n^{-1/2}{\sin(nx)}$$ with uniform limit $$f_n\rightrightarrows f\equiv 0$$. Clearly, $$f'$$ is also identically zero. However, the derivatives of the sequence of functions are given by $$f'_n(x)=n^{1/2}\cos nx,$$ and the sequence $$f'_n$$ does not converge to $$f',$$ or even to any function at all. In order to ensure a connection between the limit of a sequence of differentiable functions and the limit of the sequence of derivatives, the uniform convergence of the sequence of derivatives plus the convergence of the sequence of functions at at least one point is required:

 * If $$(f_n)$$ is a sequence of differentiable functions on $$[a,b]$$ such that $$\lim_{n\to\infty} f_n(x_0)$$ exists (and is finite) for some $$x_0\in[a,b]$$ and the sequence $$(f'_n)$$ converges uniformly on $$[a,b]$$, then $$f_n$$ converges uniformly to a function $$f$$ on $$[a,b]$$, and $$ f'(x) = \lim_{n\to \infty} f'_n(x)$$ for $$x \in [a, b]$$.

To integrability
Similarly, one often wants to exchange integrals and limit processes. For the Riemann integral, this can be done if uniform convergence is assumed:
 * If $$(f_n)_{n=1}^\infty$$ is a sequence of Riemann integrable functions defined on a compact interval $$I$$ which uniformly converge with limit $$ f$$, then $$ f$$ is Riemann integrable and its integral can be computed as the limit of the integrals of the $$ f_n$$: $$\int_I f = \lim_{n\to\infty}\int_I f_n.$$

In fact, for a uniformly convergent family of bounded functions on an interval, the upper and lower Riemann integrals converge to the upper and lower Riemann integrals of the limit function. This follows because, for n sufficiently large, the graph of $$f_n$$ is within $ε$ of the graph of f, and so the upper sum and lower sum of $$f_n$$ are each within $$\varepsilon |I|$$ of the value of the upper and lower sums of $$f$$, respectively.

Much stronger theorems in this respect, which require not much more than pointwise convergence, can be obtained if one abandons the Riemann integral and uses the Lebesgue integral instead.

To analyticity
Using Morera's Theorem, one can show that if a sequence of analytic functions converges uniformly in a region S of the complex plane, then the limit is analytic in S. This example demonstrates that complex functions are more well-behaved than real functions, since the uniform limit of analytic functions on a real interval need not even be differentiable (see Weierstrass function).

To series
We say that $\sum_{n=1}^\infty f_n$ converges:

1. pointwise on E if and only if the sequence of partial sums $s_n(x)=\sum_{j=1}^{n} f_j(x)$ converges for every $x\in E$.

2. uniformly on E if and only if sn converges uniformly as $n\to\infty$.

3. absolutely on E if and only if $\sum_{n=1}^\infty

4. f_n

5. $ converges for every $x \in E$.

With this definition comes the following result: "Let x0 be contained in the set E and each fn be continuous at x0. If $ f = \sum_{n=1}^\infty f_n$ converges uniformly on E then f is continuous at x0 in E. Suppose that $E = [a, b]$ and each fn is integrable on E. If $\sum_{n=1}^\infty f_n$ converges uniformly on E then f is integrable on E and the series of integrals of fn is equal to integral of the series of fn."

Almost uniform convergence
If the domain of the functions is a measure space E then the related notion of almost uniform convergence can be defined. We say a sequence of functions $$(f_n)$$ converges almost uniformly on E if for every $$\delta > 0$$ there exists a measurable set $$E_\delta$$ with measure less than $$\delta$$ such that the sequence of functions $$(f_n)$$ converges uniformly on $$E \setminus E_\delta$$. In other words, almost uniform convergence means there are sets of arbitrarily small measure for which the sequence of functions converges uniformly on their complement.

Note that almost uniform convergence of a sequence does not mean that the sequence converges uniformly almost everywhere as might be inferred from the name. However, Egorov's theorem does guarantee that on a finite measure space, a sequence of functions that converges almost everywhere also converges almost uniformly on the same set.

Almost uniform convergence implies almost everywhere convergence and convergence in measure.

title: Uniform convergence


text:
Talk:Render farm

kudos
Great job 213.253.39.xxx! You turned a pitiful stub into a great little article with all the essentials. --maveric149, Sunday, April 28, 2002

why it's called a render “farm”
209.19.96.75 (Talk | contribs) added this: By the way, know why it’s called a render “farm”? When Pixar first wrote Renderman, they deployed it on a few computers that were all in the same room (big computers, small room). An imaginative engineer added a line of code to the program that would have the computer play a sound when a frame completed. Someone had already host-named all the computers in the room after different animals, so the logical conclusion was to have the sound played be the sound from that animal. They could listen to the sounds and tell which computers were completing frames and which ones were hung. Hence the render farm was born.
 * I could not verify this on the internet. I would love it if it were true though, that would be awesome. As they say, "Please make sure your changes are are based on verifiable sources." —Wikibarista 05:16, 2 December 2005 (UTC)
 * Emailed Pixar for verification --Pmkpmk 12:54, 4 January 2006 (UTC)
 * Did they get back to you? I think I heard something like this before, but I suppose it's somekind of computing urban legend. There's no speaker on rendering computers. It'd be nice to have this debunked (or verified!!)''F3-R4'' 22:27, 18 September 2006 (UTC)
 * First, I'm pretty sure "render farm" is derived from the term "server farm", which predates the render farm. Second, a render farm contains a lot of computers, way more than the number of typical barnyard animals. Thirdly, even if you could come up with enough barnyard animals to name an entire render farm, now some engineer needs to go and find sound files for each and every one? No, I don't think so! --24.86.245.68 (talk) 04:48, 2 October 2013 (UTC)

Commercial application comments
Render Queue, commercial applications, common troubles., etc, could be added to this article. —Wikibarista 23:53, 3 January 2006 (UTC)
 * Do all articles need to be long? I'll see what I can do... --Pmkpmk 12:54, 4 January 2006 (UTC)
 * Oddly, any attempt I've made to add references to commercial applications has been met with a "no spam" comment by others. Granted, I've included my own commercial application each time, but have not removed others' when they were present and on my most recent attempt included some other commercial applications (render queues and the like) in an attempt to keep things non-biased.  So what is the actual Wikipedia policy regarding linking to commercial sites, and where is that policy written? --Early Ehlinger [ResPower] (previous comment posted from 24.214.236.20)
 * Hi Early, welcome to Wikipedia. The commercial applications I was referring to were not software programs, but the application of a render farm in a commercial environment. Who uses them, and how.
 * To answer your question anyway, Wikipedia doesn't have strict policies, but policies and guidelines if you want to better understand. Go there and look around. As for why many of us don't feel that it's appropriate to put a link to a web site for a business, I think it's because it simply advertises a service and subsequently some other products, with no real context. There are many, many, many products involved in this render queues, and it must not seem appropriate to us that it be featured in the article. It doesn't help with the explanation of a render queue.
 * There are many more people who can write all this a lot more eloquently, and I'm sure they have, so I'd instead encourage you to read those policies and guidelines, the vanity guidelines (especially the section Vanity vs: encyclopedic, External link spamming, and what Wikipedia is not. If you still feel that it is important to use Wikipedia for your link, then there is always the larger community to determine whether or not it should be included here. Sorry if it sounds so pedantic, I'm trying my best to explain it all. Please consider becoming a Wikipedian and creating an account. See this welcome message —Wikibarista 06:08, 3 March 2006 (UTC)
 * Hi Wikibarista, Thanks so much for explaning things and giving good pointers to the detailed policies involved. I think part of what bothers me is that there appears to be a bias in the Wikipedia against any mention of commercial services.  BURP, for example, is simply a non-commercial alternative to services like the one that I offer, and yet its presence on this page and indeed in Wikipedia as a whole goes completely unchallenged.  Does its presence here really offer anything that a discussion about commercial render farms does not? -- Early Ehlinger, 14 March 2006 —The preceding comment was added by Earlye (talk • contribs) 18:55, 14 March 2006 (UTC) Please sign your posts with ~

 * I have removed the links to Rebus for the same reason that others have removed the links to ResPower. Not to be petty, but if links to my service should not be allowed, then links to other services likewise should be removed.  -- Early Ehlinger. —The preceding comment was added by Earlye 27 June 2007

 * I have removed the links to render-it for the same reason that others have removed the links to ResPower. Not to be petty, but if links to my service should not be allowed, then links to other services likewise should be removed.  -- Early Ehlinger. —The preceding comment was added by Earlye 2 Jan 2009  —Preceding undated comment was added at 18:32, 2 January 2009 (UTC).

Dr Queue

 * Would drqueue fit here ? drqueue.org . It was used for elephants dream and it's open source. Thanks. —The preceding unsigned comment was added by 87.216.187.247 (talk) 6 September 2006.

 * I think there's a fine line between spamming and citing notable softwares. Like in this article, renderman (PRman) should be cited because it is the de facto leader in the entertainment industry, right? Along with mental ray to a lesser extent.''F3-R4'' 22:19, 18 September 2006 (UTC)

Gross generalization?
"As a rule of thumb, CG images take roughly an hour per frame to render."

At what quality? what resolution? how detailed is the scene? this line is completely fallacious in its current context, rendering a 100x100 picture of a square will take under a second. Rendering hugely complex scenes of massive size could take many hours or even days. Also entirely dependent on the CPU speed of the computer and how many computers in the render farm. I'm assuming that this is meant to be a reference to how long it takes for CG movie projects like shrek. This could be completely different for making CG movies for computer games, which are often made on lower budget and may have smaller Render Farms. It should give specific details. Something like "X Company, making Y movie had Z computers in a render farm which took roughly an hour per frame to render"

 * To say, render a 100x100 frame takes a second, well, to render a cube with only one light and no ray tracing or shadow or whatever else there may be at 1080i would take about one second. Frame size is less of an indicator of rendering speed as to what is getting rendered. (deformations, displacements, particles, clothes, fluids etc) As the article refers to render farms, it should relate to the time it takes to render the stuff they usually render on render farms. On the Simpsons season 7 DVD (to cite a source) the PDI guy said that the rule of thumb should be "rendering time stays constant" it still takes hours to render a frame as it did in the 90s but now a frame has more element to render. Computing power has increased, but also the complexity of the software that runs on them, thus the rendering time stays constant.(PDI is the folks who would now become dreamworks animation who made shrek etc) But then, pixar in one of them press conference cited something like average 8 hours. The average time to render something on a render farm depends on what software is being run. A mental ray rendering a certain frame wouldn't yield an equivalent result as it would if that certain frame is rendered on a PRman. In summary, I thought that phrase needs to be removed and replaced with a more "wikipedia" explanation.''F3-R4'' 22:37, 18 September 2006 (UTC)

 * Did my best to rewrite it but keep the sentiment. Originally, it was two lines long, so I may have lengthened it to be a little to repetitive, feel free to reword the last two sentences a bit. —Wikibarista 16:37, 19 September 2006 (UTC)

Example of time
Can someone put examples of some movies and how long it took to render them. Like "Movie X took Y months to render". —Preceding unsigned comment added by Frap (talk • contribs) 18:47, 20 January 2008 (UTC)

Too Technical
I have to agree, this article is a bit too technical. It didn't help me better understand what this or how it works at all. Any chance this will be changed in the near future? ——Digital Jedi Master (talk) 05:19, 8 June 2010 (UTC)

And also an incorrect article

 * Well, lucky you, Digital Jedi Master. You did not absorb the many errors therein. Zero references (except a dictionary item!) of course, and the entire process description is at times outdated and more often incorrect. I may get to fix it one day.... For now I just tagged it to avoid error propagation. History2007 (talk) 14:42, 1 February 2012 (UTC)

title: Talk:Render farm