title: Madrid Metro

text: "MetroSur", one of the largest ever civil engineering projects in Europe, opened on 11 April 2003. It includes of tunnel and 28 new stations, with a new interchange station on Line 10, connecting it to the city centre and stations linking to the local train network. Its construction began in June 2000 and the whole loop was completed in less than three years. It connects Getafe, Móstoles, Alcorcón, Fuenlabrada, and Leganés, five towns located in the area south of Madrid.


title: Madrid Metro

text: Most of the efforts of Madrid regional government in 2000s were channeled towards the enlargement of the Metro network. In the 2003–2007 term, President Esperanza Aguirre funded a multibillion-euro project, which added new lines, and joined or extended almost all of the existing metro lines. The project included the addition of of railway and the construction of 80 new stations. It brought stations to many districts that had never previously had Metro service ("Villaverde, Manoteras, Carabanchel Alto, La Elipa, Pinar de Chamartín") and to the eastern and northern outskirts as well ("Coslada, San Fernando de Henares, Alcobendas, San Sebastián de los Reyes"). For the first time in Madrid, three interurban light rails ("Metro Ligero" or ML) lines were built to the western outskirts ("Pozuelo de Alarcón, Boadilla del Monte") - mL2 and mL3 - and to the new northern districts of Sanchinarro and Las Tablas - mL1. As a last minute addition, a project on line 8 connected it to the new T4 terminal of Madrid-Barajas Airport.


title: Madrid Metro

text: Many pending expansion projects were stalled by the economic fallout of the 2008 financial crisis. In recent years, improving the existing network has taken priority over extending it, with major projects like the Plan de Accesibilidad () being put into place to enhance accessibility for those with physical disabilities. Currently, there are more than 530 elevators in the system, a number only set to grow with the advancement of the accessibility plan. Additionally, by the end of 2020, the entire network will have gapless cell phone coverage.


title: Madrid Metro

text: As both the number of trips in the Metro and the number of inhabitants of the Comunidad de Madrid have seen steady increases in recent years, there has been a growing political consensus that further extensions are necessary, specifically to support the single central circular Line 6 in a system of mostly radial lines.


title: Madrid Metro

text: Line 11 will be extended by 6.3 kilometers from Plaza Elíptica towards Conde de Casal, both Line 6. Construction is set to start in 2020, with a new bus line interchange to be built at Conde de Casal. The extension will provide access to Lines 1 and 3 as well as Atocha railway station, with a new station Madrid Río to be constructed in the vicinity of the Manzanares.Revised due to Coivd-19, Line 11 is presently planned to terminate near the airport, as opposed ton the plan in 2007 to Avenida de la Ilustración. The planned extension would start construction in 2024 and be operational in 2030.


title: Madrid Metro

text: Under the new regional government, other long-stalled expansion projects have also gained new momentum. This includes the extension of Line 5 to Barajas airport, as well as a connection between Line 3 and MetroSur in the south.


title: Madrid Metro

text: In the context of the Madrid Nuevo Norte project (previously Operación Chamartín), there are plans for the construction of a new metro line from the Chamartín railway terminal with three stations.


title: Madrid Metro

text: A further extension of the Line 11 will extend from Conde de Casal to Valdebebas Norte, which will link to 2 new tunnels at Pueblo Nuevo and Mar de Cristal.


title: Madrid Metro

text: The age of many Madrid Metro stations is evident by their design: Older stations on the narrow lines are often quite compact, similar to stations of the Paris Metro. They were decorated with tilings in different colour schemes depending on the station. In recent years, most of these stations have been refurbished with single-coloured plates matching those in the newest ones, alongside other improvements such as modernized signalling technology and elevators. The stations built between the late 70s and the early 90s are slightly more spacious, with most of them having cream-coloured walls.


title: Madrid Metro

text: On the other hand, the most recent stations are built with space in mind, and have natural-like lighting and ample entryways. The colour scheme varies between stations, using single-colored plates and covering the whole station in light colors. Recently built transfer stations have white walls, but this is not the norm.


title: Madrid Metro

text: Most stations are built with two side platforms, but a handful of them (the busiest transfers) have a central island platform in addition to the side platforms theoretically dedicated to exits. This system was originally used on the Barcelona Metro and is called the Spanish solution. The 12 stations with this setup are:


title: Madrid Metro

text: Two stations have cross-platform interchange arrangement with two island platforms, which allows extremely fast transfers between lines. Both of these stations are on Line 10, with cross-platform interchanges at Príncipe Pío (with ) and Casa de Campo (with ). On both occasions, Line 10 uses the outside tracks, so passengers unboarding there leave through the "right" side of the train instead of the usual left side.


title: Madrid Metro

text: In addition, 10 stations are built with just one island platform instead of the usual side platforms. These stations are:


title: Madrid Metro

text: Another system is where there is one island platform with one side platform. This system is used in two stations on lines 2 and 4 as termini, and three stations on Lines 7, 9, and 10 where it is required for passengers to change to smaller trains to continue their journeys, normally to towns outside Madrid like Alcobendas or Coslada. This is done so the island platform can be used for passengers to change easily between trains. In the latter three stations, the island platform is equipped with fare gates in order for tickets to be validated for travel between fare zones. These stations are:


title: Madrid Metro

text: Since 1999 Metro de Madrid has used a patented system for its installations: a solid rail hung from the ceiling of the tunnels, instead of the usual copper or aluminium wire hung from overhead gantries at regular intervals. This type of overhead line is rigid, making it more robust and less prone to failures. Installations outside of tunnels are rare, as they require many more support structures compared to traditional wire based overhead lines, making them more expensive to install. This system of rigid overhead power supply is also used in other metro systems.


title: Madrid Metro

text: The Metro conventional network has 242 stations on 12 lines plus one branch line, totalling , of which approximately 96% of stations are underground. The only surface parts are between Empalme and west of Eugenia de Montijo (); between Lago and north of Casa de Campo (); and between south of Puerta de Arganda and Arganda del Rey (), for a total of 8 aboveground stations. Additionally, some 30 km of Metro Ligero (light rail) lines across serve the various regions of the metropolitan area which have been deemed not populated enough to justify the extraordinary spending of new Metro lines. Combined, they have 38 stops, of which 4 also connect to the conventional Metro system. Most of the ML track length is on surface, usually running on platforms separated from normal road traffic. However, ML1 line has some underground stretches and stations.


title: Madrid Metro

text: Traditionally, the Madrid metro was restricted to the city proper, but today nearly one third of its track length runs outside the border of the Madrid municipality. Today, the Metro network is divided in six regions:


title: Madrid Metro

text: At most of the borders between the regions, one has to switch trains even when staying in the same line, because the train frequency is higher in the core MetroMadrid than in the outer regions.


title: Madrid Metro

text: Madrid also has an extensive commuter train ("Cercanías") network operated by Renfe, the national rail line, which is intermodal with the metro network. In fact, 22 Cercanías stations have connections to the Metro network, which is indicated on the official map by the Cercanías logo. Many of the new lines since 1999 have been built to link to or end at Cercanías stations, like the ML2 line, which ends at the Aravaca station providing a fast entry into Madrid though the C-7 or C-10 commuter lines and arriving in only one step to the bus and Metro hub Príncipe Pío ( ).


title: Madrid Metro

text: Traditionally, the trains operating in the Madrid Metro have been built and supplied by the Spanish company Construcciones y Auxiliar de Ferrocarriles (CAF). This was particularly true under Francisco Franco's dictatorship, due to the politic of autarky his administration initially pursued. However, despite CAF still working for the Metro, in recent years the Italian AnsaldoBreda has also provided trains for the wide-profile lines.


title: Madrid Metro

text: Every rolling unit in the Madrid Metro has a unique ID that singles it out in the whole network. Those IDs are grouped by the rolling unit model (the "series") and thus is used to categorize the trains, as they bear no user-visible statement of the model specified by the manufacturer. An ID is made up of:


title: Madrid Metro

text: CAF series 2000: This series has two separate sub-series usually called A and B. The first batch, while reliable and practical, was extremely "box-like" in its looks. They are nicknamed 'Pandas', after a car by Seat with the same name and similar boxy design. In contrast, the B sub-series train sets can be told apart by its sleeker, rounder forms, which has granted them the nickname of "bubble" ("") for their round driver cabin window. Series 2000A are currently the more numerous in the network: 530 cars were built and delivered between 1985 and 1993, having serviced every narrow profile line. They are also among the oldest stock in operation in the Madrid Metro. The most reliable ones are being refurbished and painted with new, lighter colors like the ones used in Series 3000. Series 2000B were delivered in lesser numbers (about 126 cars) between 1997 and 1998, with the inclusion of air conditioning and station announcements through pre-recorded voice messages and LED displays. They are currently used in line 5, with no plans for retirement.


title: Madrid Metro

text: CAF series 3000: The newest of the narrow line trainsets, series 3000 were commissioned for the reopening of line 3 after its complete renewal in the early 2000s. Their constituent subunits can be completely joined through crossable articulations, making it possible to go from the head to the tail without actually exiting the train. This has earned them the nickname of "boa", a term usually applied in Spain to double-length buses with such joints. They are currently servicing lines 2, R, 3, 4 and 5. Series 3000 trains look rather like a narrowed version of series 8000, while the interior uses mainly yellow and light blue tones.


title: Madrid Metro

text: CAF series 5000: Currently servicing line 9 and occasionally line 6, this model has had a long history: the first trainsets were delivered in 1974 for the newly opened, first wide-profile line 7, while the latest subseries, 5500, of which 24 trainsets of 6 cars each were built, entered service in 1993. They were the last to use the old, square "box-like" design from CAF, which was already becoming unpopular for its exaggerate priming of effectiveness versus aesthetics. The first iteration featured a wood lookalike coating for the inner walls and a novel seat distribution in two-seat rows perpendicular to the train walls, making them look not unlike older regional trains. Subseries 5100-5200 returned to the traditional seating along the train walls, but still included another feature from the first iteration, automatic opening of all the gates in the train. The final subseries, 5500, has a distinct, darker color scheme and returns to the usual on-demand opening of train gates with a button on each one. Being the oldest rolling stock in operation in the wide profile lines, many cars were retired or sold to the Buenos Aires Underground for operation on Line B to make up for shortfalls on the line following extensions.


title: Madrid Metro

text: CAF series 6000: This model, of which 29 trainsets were built and delivered in 1998, was the first by CAF to feature a new, sleeker and rounder design. As it was to serve TFM, the stretch of line 9 connecting Madrid to Arganda del Rey (the first extension of the Metro network outside Madrid proper), its interior resembles the regional Cercanías trains more closely than any other Metro trains: compact seats in couples set perpendicularly to the train walls, more places to grasp in case of a sudden brake/acceleration, etc. They were also the first to include luminous panels stating their destination, as the line they service was effectively split in two stretches, and travellers had to switch trains at Puerta de Arganda. Finally, they primed the "boa train" layout, but the walkable aisle only spanned two cars, while a trainset would usually carry 4 or 6. These trains are equipped with automatic train protection (ATP) and automatic train operation (ATO). Series 6000 is currently doing service on line 9b (TFM). In 2013, 73 of the 108 cars ordered were sold to Buenos Aires for operation on Line B of the metro system; the sale totalled €32.6 million for the retirement of Japanese-built units, with a further 13 cars ordered at a later date. These trains have been widely criticised in Argentina, and been called the worst purchase in the history of the Buenos Aires Underground.


title: Madrid Metro

text: Ansaldobreda series 7000 & 9000: The first purchase to a manufacturer other than CAF, and to a non-Spanish dealer, 37 series 7000 trainsets service the extremely busy line 10. They were the first in the network to feature a full "boa" layout, allowing commuters to traverse the whole six cars. They are extremely functional, with ample 1.3m doors and a sleek, unobtrusive design for a total capacity of 1,260 people per trainset (180 seated). This model also features two TV screens in each car, but they are left unused, both regularly or in emergencies. Series 9000 trains are similar to their previous incarnation, but include better accessibility for disabled people and more safety measures, such as visual and auditive warnings for the train gates and more effective emergency brakes, they also brought small aesthetic changes like the removal of the wood effect from the ceiling and the change of the red top stripe of the doors to a blue color. Series 7000 currently service the main part of line 10 from Puerta del Sur to Tres Olivos and occasionally on line 9; while series 9000 comprise the main fleet of line 7 and 12, occasionally on line 10, and on line 9 to cover for the sold 6000s.


title: Madrid Metro

text: CAF series 8000: Originally designed for the MetroSur line 12, 45 trainsets were built and delivered by CAF in 2002. Each one is composed of three cars or four cars joined in the "boa" layout, with the three car version servicing line 12 and the four car version servicing line 8 as-is, while line 9 & line 10 services use two such trainsets to form a MRM-MRM configuration for a maximum of 1,070 passengers (144 seated). The interior distribution is rather like that of series 7000, with a bigger clear area (i.e. without seating) in the first car for people carrying luggage to/from the airport and disabled people in wheelchairs. Like the narrower series 3000 trainsets, its bogies are insonorized and feature a hybrid rubber-pneumatic suspension system. Series 8000 primed the introduction of regenerative braking in the Madrid Metro. The system reverses the normal circuit of the electric motors when braking, thus making the deceleration "return power" to the network. Also, they feature the now-standard informative panels and gate activity warnings in the interior. A second batch was ordered for line 11 to replace the series 3000 operating on the line since the extension of the line to La Fortuna in 2010. The original batch currently services lines 12 and 8, while also providing rush hour support to lines 9 and 10 whilst the second batch currently services line 11.


title: Madrid Metro

text: CAF series 8400: Derived from the recent series 8000 trains, the 8400 series are the newest train type to enter service on the Madrid Metro on line 6 since 2010 to complement the older series 5000 serving on that line. It currently services line 6.


title: Madrid Metro

text: Alstom Citadis 302: The vehicles serving the light rail lines are low-floor articulated trams in a five-section "boa" configuration, which allows for a maximum of about 200 passengers per tram (60 seated). They can reach a top speed of 100 km/h (65 mph), but in practice they are limited to 70 km/h (45 mph) in most track stretches, and even less in urban sprawls. The tram features a bell-like proximity warning that is activated when the train approaches a station or a level crossing with pedestrians, which has given rise to complaints from people living near the tracks because of the noise generated. Safety features also include door activity warnings for passengers and emergency brakes comparatively more effective than in any other train dedicated to Metro service, as the trams, though remaining in their own lanes separated from other traffic, can cross roads and populated areas.


title: Madrid Metro

text: Currently, Metro Ligero has four lines, although one of them is located outside of the city of Madrid in its entirety:


title: Madrid Metro

text: Until the early 1990s and the transfer of the Metro system to the Autonomous Community of Madrid, the rate of investment in the network by the central government was extremely low, and thus very old trains were used way beyond their intended lifespans. Particularly loathed was the case of line 5, which was serviced by the nearly 40-year-old series 300 and 1000 from CAF. It was not uncommon that a child would ride to school on the same train his/her parents took decades earlier. Some renewals, along with the purchases of series 2000A and 5000, were started by the socialist regional government of Joaquín Leguina, but in 1995 the People's Party took over the government with the promise to widely extend and improve the Metro service. New lines were built and old ones refurbished: line 5 service was disturbed for several years as some stations at a time were closed and refitted, while line 3 was closed for two consecutive summers in order to expand its platforms to 90 m. Then, new rolling stock was also requested: 1998 saw the arrival of the first CAF series 2000B, retiring the infamous series 1000. Initially the better-preserved series 300 were refitted and painted in the new blue-white color scheme (from the old red corporate image), but they were also retired with the arrival of more series 2000B and, finally, series 3000.


title: Madrid Metro

text: The Madrid Metro network is split into the six "functional" zones mentioned above. Each one has a "single" ticket (Billete Sencillo), valid for one trip within the zone, and a 10-trip ticket for a comparatively lower price. When crossing zone boundaries, one has to buy a new ticket for the zone being entered. There is also a "combined" ticket, which provides for a single trip between any two points of the network except the Airport stations, which have an additional supplement of €3. All in all, it is possible to go from the airport to any other point of the network for up to €5.00. All tickets are issued on the Tarjeta Transporte Público.


title: Madrid Metro

text: Also, the Consorcio Regional de Transportes (Regional Transportation Authority) has a division of its own, with geographic zones named A through C2. This body sells monthly and annual passes for unlimited trips within their zone of validity, and also a range of Tourist Passes for 1, 3, 5 or 7 days. All of them are accepted at the Metro stations within their zones, and passengers using a CRT pass do not have to pay the airport supplement.


title: Madrid Metro

text: The metro is operated by its own company, under the Department of Public Works, City Planning, and Transportation of the autonomous community of Madrid. The passage between Puerta de Arganda (Line 9) and Arganda del Rey (Line 9) is operated by "Transportes Ferroviarios de Madrid" (TFM), while the second and third lines of Metro Ligero are operated by "Metro Ligero Oeste" (MLO). All of Madrid's rapid transit systems are members of the Consorcio Regional de Transportes, which sells monthly passes for unlimited use of the metro, bus and commuter train networks within the area covered by the pass.


title: Madrid Metro

text: Some underground stations are large enough to hold public events, such as the three-day fitness festival in May 2011, which attracted 2,600 visitors. Ópera station contains a 200-square-metre archaeological museum.


title: Madrid Metro

text: Various metro stations show contemporary art. The exhibition "100 años de Metro" (100 years of Metro) has received more than 27,000 visitors. At Expometro within the Retiro station, multiple exhibitions of modern art have been on display, such as the exhibition by Pablo Sycet, Rafael Arellano, Tono Carbajo, Christian Domec and Julio Juste called "The Dream of Madrid", 1986, or ""The Passengers"" by Daniel Garbade (2000). Both platforms of the station exhibit murals by Antonio Mingote. The Goya station shows works by Francisco Goya (1746–1828) on two murals on the platforms of line 2.


title: Savitr

text: Savitṛ (Sanskrit: stem ', nominative singular '), also rendered as Savitur, in Vedic scriptures is an Aditya i.e. off-spring of the Vedic primeval mother goddess Aditi. His name in Vedic Sanskrit connotes "impeller, rouser, vivifier."


title: Savitr

text: He is sometimes identified with—and at other times distinguished from—Surya, "the Sun god". When considered distinct from the Sun proper, he is conceived of as the divine influence or vivifying power of the Sun. The Sun before sunrise is called Savitr, and after sunrise until sunset it is called Sūrya.


title: Savitr

text: Savitr is venerated in the Rig Veda, the oldest component of the Vedic scriptures. He is first recorded in book three of the Rigveda; (RV 3.62.10) later called the Gayatri mantra. Furthermore, he is described with great detail in Hymn 35 of the Rig Veda, also called the Hymn of Savitr. In this hymn, Savitr is personified and represented as a patron deity. He is celebrated in eleven whole hymns of the Rig Veda and in parts of many others texts, with his name being mentioned about 170 times in aggregate.


title: Savitr

text: Savitr disappeared as an independent deity from the Hindu pantheon after the end of the Vedic period, but is still worshiped in modern Hinduism and is referred to as "Sāvitrī."


title: Savitr

text: Savitr is a deity whose name primarily denotes an agent, in the form of a noun derived from a verbal root with the agent suffix "-tṛ" added.


title: Savitr

text: The name of Savitr belongs to a class of Vedic theonyms, together with Dhatṛ, Tratṛ and Tvastr. These names denote that these are agent gods, who create, protect, and produce, respectively.


title: Savitr

text: Savitr has golden arms, and is broad-handed or beautiful-handed. He is also pleasant tongued or beautiful-tongued, and is once called iron-jawed. His eyes are golden as well. He is yellow-haired, an attribute shared with Agni and Indra. He dons on a tawny garb. He has a golden chariot with a golden axle, which is omni-form, just as he himself is capable of assuming all forms. His channel is analogized as a resplendent chariot drawn by two radiant steeds or by two or more bronze, white-footed stallions. Mighty splendour ("amati") is preeminently attributed to Savitr, and mighty "golden" splendour to him only. Such splendour he stretches out or diffuses. He illumines the air, heaven and earth, the world, the spaces of the earth, the vault of heaven.


title: Savitr

text: Like Pushan and Surya, he is lord of that which is mobile and is stationary. Savitr has been attributed to as upholding the movables and immovable, which signifies the maintenance of Ṛta. Savitr is a beneficent god who acts as protector of all beings, who are provident and guard the world of spirits. Being an Aditya, Savitr is true to the Eternal Order and act as the score exacter.


title: Savitr

text: His primordial pathways in the air are dustless and sleekly traversed, on them he is besought to fortify his invokers. He is prayed to convey the departed soul to where the righteous dwell. Savitr bestows immortality on the gods as well as length of life on man. He also bestowed immortality on the Rbhus, who by the greatness of their deeds advanced to his dwelling. Like other gods, Savitr is a supporter of the cosmos. Also, he holds the whole world, a role which was also assigned to Vishnu in the Vedas.


title: Savitr

text: The class, judged by the evolution of the mythological creations of the Veda, does not represent direct abstractions, but appears in each case to be derived from an epithet applied to one or more deities, illustrating a particular aspect of activity or character. Such epithets gradually become detached, finally attaining an independent status. Thus "Rohita", the 'Red One' (whose female form is "Rohinī"), originally an epithet of the sun, as a separate deity in the capacity of a Creator.


title: Savitr

text: According to Yaska, Sanskrit scholar of the 5th century BCE, who made various attempts to interpret difficult Vedic mythologies in his work "Nirukta" ("Etymology") (12, 12), the time of Savitr’s appearance is when darkness has been removed. Sayanacharya (on "Rig Veda") remarks that before his rising the sun is called Savitr, but from his rising to his setting, Surya. But Savitr is also sometimes spoken of as "sending to sleep", and must therefore be connected with evening as well as morning. He is, indeed, extolled as the setting sun in one hymn (2, 38); and there are indications that most of the hymns addressed to him are meant for either a morning or an evening sacrifice. He brings all two-footed and four-footed beings to rest and awakens them. He unyokes his steeds, brings the wanderer to rest; at his command night comes; the weaver rolls up her web and the skilful man lays down his unfinished work. Later the west was wont to be assigned to him, as the east to Agni, and the south to Soma.


title: Savitr

text: Like Surya, Savitr is implored to remove evil nightmares and to render men sinless. Savitr drives away sorcerers and antagonism. He observes fixed laws. The waters and the wind are subject to his ordinance. He leads the waters and by his propulsion they flow broadly. The other gods follow his lead. No being, not even Indra, Varuna, Mitra, Aryaman, Rudra, can resist his will and independent dominion. His praises are celebrated by the Vasus, Aditi, Varuna, Mitra and Aryaman. He is lord of all things worthy, and bestows blessings pervading heaven, air, earth.


title: Savitr

text: ... the connection of Savitr with the sun is fairly close. It is at least possible, therefore, that in its origin Savitr was not an independent creation, but was an epithet of Surya, but that question is of little importance: The essential feature of the god is not his original basis, but his function as the inspirer or impeller to holy sacrifice: The ritual act is repeatedly said in the Yajur Veda to be done ‘on the instigation of the god Savitr’.


title: Savitr

text: In several passages of the "Rig Veda", Savitr and Surya appear simultaneously. It may even appear based on A.B. Keith's opinion that the terms Savitar and Surya are used interchangeably in certain hymns of the "Rig Veda". However it is worth noting that several other deities are directly associated with the epithet of Savitar in the "Family Books". They include Indra who is paired alongside Savitar, and Tvastr who is compounded with Savitar. Furthermore, Savitar is unambiguously identified with Bhaga. Savitar is also unambiguously called Pusan and Mitra. While Savitar certainly has directly been charged with using Surya's rays, Savitar has a much more direct congruence with other deities. The Vedic poet observes:


title: Savitr

text: In another hymn Surya is spoken of in terms "Prasavitṛ" (Vivifier), an adjective usually applied to Savitr, and in the third verse Savitr is apparently mentioned as the same god as Surya. In other hymns also, it is hardly possible to separate the two deities. In certain passages, Savitr combines with the rays of the sun or shines with the rays of the sun.


title: Savitr

text: "Indra measured six broad spaces, from which no existing thing is excluded: He it is who made the wide expanse of earth and the lofty dome of the sky, even he." Savitr assisted Indra in shaping the universe.


title: Savitr

text: [T]here are in the last book of the Rig Veda some hymns which treat the origin of the world philosophically rather than mythologically. Various passages show that in the cosmological speculation of the Rig Veda The sun was regarded as an important agent of generation. Thus he is called the soul (atma) of all that moves and stands. Statements such as that he is called by many names though one indicate that his nature was being tentatively abstracted to that of a supreme god, nearly approaching that of the later conception of Brahma. In this sense the sun is once glorified as a great power of the universe under the name of the golden embryo, hiranya-garbha, in "Rig Veda". It is he who measures out space in the air and shines where the sun rises. In the last verse of this hymn, he is called Prajapati, lord of created beings , the name which became that of the chief god of the Brahmanas. It is significant that in the only older passage of the Rig veda in which it occurs, Prajapati is an epithet of the solar deity Savitr, who in the same hymn is said to rule over what moves and stands.


title: Savitr

text: The Vedas do not specifically identify the Ādityas as there is no classification of the thirty-three gods, except for in the Yajurveda (7.19), which says that there are eleven gods in heaven (light space), eleven gods in the atmosphere (intermediate space), and eleven gods on earth (observer space). In some passages of the Satapatha Brahmana, the number of Ādityas is eight, and in other passages twelve Ādityas are mentioned.


title: Savitr

text: Some modern Hindu spiritual thinkers assign symbolism to the Vedic deities like Savitr. The Vedic deities are not only forces of nature, but also forces that exist within the human intellect and psyche, and help the individual in spiritual progress.


title: Savitr

text: According to Sri Aurobindo, the Vedic depictions are deeper than mere imagery. The gods, goddesses and the evil forces mentioned in the Vedas represent various cosmic powers. They play a significant role in the drama of creation, preservation, and destruction in the inner world of a human being.


title: Savitr

text: Once the senses are controlled and the mind is stabilized through slaying of all the dark powers, comes the awakening, the goddess Ushas, who brings along with her Ashvins into the world of inner consciousness. After Ushas appears Aditi, the Primal Sun, the God of Light: First as Savitr, who represents the divine grace essential for all spiritual success, and then as Mitra, who as the divine love is considered as a friend of the illumined mind (Indra) and his associates (the other gods). The Sun is of Truth, after which appear Rta (Truth in Action) and Rtachit (Truth consciousness).


title: Savitr

text: In DC Comics' The Flash comics and The CW's "The Flash" TV series, the speedster Savitar is an enemy of the Flash who named himself after the Hindu god.


title: Savitr

text: In a fiction by author Ryan Sequeira, called "EvOLv", where Savitr has been named as one of the Supreme Gods - parallel with Shiva - The God Savitr is referred to as the source of light in the multiverse.


title: Savitr

text: In the Dark-Hunter fantasy series by author Sherrilyn Kenyon, Savitar is a Chthonian god killer who is thousands of years old and was responsible for policing the Atlantean pantheon.


title: Delta method

text: In statistics, the delta method is a result concerning the approximate probability distribution for a function of an asymptotically normal statistical estimator from knowledge of the limiting variance of that estimator.


title: Delta method

text: The delta method was derived from propagation of error, and the idea behind was known in the early 19th century. Its statistical application can be traced as far back as 1928 by T. L. Kelley. A formal description of the method was presented by J. L. Doob in 1935. Robert Dorfman also described a version of it in 1938.


title: Delta method

text: While the delta method generalizes easily to a multivariate setting, careful motivation of the technique is more easily demonstrated in univariate terms. Roughly, if there is a sequence of random variables satisfying


title: Delta method

text: where "θ" and "σ" are finite valued constants and formula_2 denotes convergence in distribution, then


title: Delta method

text: Demonstration of this result is fairly straightforward under the assumption that is continuous. To begin, we use the mean value theorem (i.e.: the first order approximation of a Taylor series using Taylor's theorem):


title: Delta method

text: Note that since formula_6 and formula_7, it must be that formula_8 and since is continuous, applying the continuous mapping theorem yields


title: Delta method

text: By definition, a consistent estimator "B" converges in probability to its true value "β", and often a central limit theorem can be applied to obtain asymptotic normality:


title: Delta method

text: where "n" is the number of observations and Σ is a (symmetric positive semi-definite) covariance matrix. Suppose we want to estimate the variance of a scalar-valued function "h" of the estimator "B". Keeping only the first two terms of the Taylor series, and using vector notation for the gradient, we can estimate "h(B)" as


title: Delta method

text: One can use the mean value theorem (for real-valued functions of many variables) to see that this does not rely on taking first order approximation.


title: Delta method

text: Hence, even though for any finite "n", the variance of formula_24 does not actually exist (since "X" can be zero), the asymptotic variance of formula_25 does exist and is equal to


title: Delta method

text: Note that since "p>0", formula_27 as formula_28, so with probability converging to one, formula_29 is finite for large "n".


title: Delta method

text: Moreover, if formula_30 and formula_31 are estimates of different group rates from independent samples of sizes "n" and "m" respectively, then the logarithm of the estimated relative risk formula_32 has asymptotic variance equal to


title: Delta method

text: This is useful to construct a hypothesis test or to make a confidence interval for the relative risk.


title: Delta method

text: The delta method is often used in a form that is essentially identical to that above, but without the assumption that or "B" is asymptotically normal. Often the only context is that the variance is "small". The results then just give approximations to the means and covariances of the transformed quantities. For example, the formulae presented in Klein (1953, p. 258) are:


title: Delta method

text: When the delta method cannot be applied. However, if exists and is not zero, the second-order delta method can be applied. By the Taylor expansion, formula_35, so that the variance of formula_36 relies on up to the 4th moment of formula_37.


title: Delta method

text: The second-order delta method is also useful in conducting a more accurate approximation of formula_36's distribution when sample size is small.


title: Delta method

text: For example, when formula_37 follows the standard normal distribution, formula_36 can be approximated as the weighted sum of a standard normal and a chi-square with degree-of-freedom of 1.


title: Delta method

text: A version of the delta method exists in nonparametric statistics. Let formula_42 be an independent and identically distributed random variable with a sample of size formula_43 with an empirical distribution function formula_44, and let formula_45 be a functional. If formula_45 is Hadamard differentiable with respect to the Chebyshev metric, then


title: Delta method

text: where formula_48 and formula_49, with formula_50 denoting the empirical influence function for formula_45. A nonparametric formula_52 pointwise asymptotic confidence interval for formula_53 is therefore given by


title: Delta method

text: where formula_55 denotes the formula_56-quantile of the standard normal. See Wasserman (2006) p. 19f. for details and examples.


title: Discovery of the neutron

text: The discovery of the neutron and its properties was central to the extraordinary developments in atomic physics in the first half of the 20th century. Early in the century, Ernest Rutherford developed a crude model of the atom, based on the gold foil experiment of Hans Geiger and Ernest Marsden. In this model, atoms had their mass and positive electric charge concentrated in a very small nucleus. By 1920 chemical isotopes had been discovered, the atomic masses had been determined to be (approximately) integer multiples of the mass of the hydrogen atom, and the atomic number had been identified as the charge on the nucleus. Throughout the 1920s, the nucleus was viewed as composed of combinations of protons and electrons, the two elementary particles known at the time, but that model presented several experimental and theoretical contradictions.


title: Discovery of the neutron

text: The essential nature of the atomic nucleus was established with the discovery of the neutron by James Chadwick in 1932 and the determination that it was a new elementary particle, distinct from the proton.


title: Discovery of the neutron

text: The uncharged neutron was immediately exploited as a new means to probe nuclear structure, leading to such discoveries as the creation of new radioactive elements by neutron irradiation (1934) and the fission of uranium atoms by neutrons (1938). The discovery of fission led to the creation of both nuclear power and nuclear weapons by the end of World War II. Both the proton and the neutron were presumed to be elementary particles until the 1960s, when they were determined to be composite particles built from quarks.


title: Discovery of the neutron

text: At the start of the 20th century, the vigorous debate as to the existence of atoms had not yet been resolved. Philosophers such as Ernst Mach and Wilhelm Ostwald denied that atoms were real, viewing them as a convenient mathematical construct, while scientists such as Arnold Sommerfeld and Ludwig Boltzmann saw that physical theories required the existence of atoms.


title: Discovery of the neutron

text: Radioactivity was discovered in 1896 by the French scientist Henri Becquerel, while working with phosphorescent materials. In 1898, Ernest Rutherford at Cavendish Laboratory distinguished two types of radioactivity, alpha rays and beta rays, which differed in their ability to penetrate, or travel into, ordinary objects or gases. Two years later, Paul Villard discovered gamma rays, which possessed even more penetrating power. These radiations were soon identified with known particles: beta rays were shown to be electrons by Walter Kaufmann in 1902; alpha rays were shown to be helium ions by Rutherford and Thomas Royds in 1907; and gamma rays were shown to be electromagnetic radiation, that is, a form of light, by Rutherford and Edward Andrade in 1914. These radiations had also been identified as emanating from atoms, hence they provided clues to processes occurring within atoms. Conversely, the radiations were also recognized as tools that could be exploited in scattering experiments to probe the interior of atoms.


title: Discovery of the neutron

text: At the University of Manchester between 1908 and 1913, Rutherford directed Hans Geiger and Ernest Marsden in a series of experiments to determine what happens when alpha particles scatter from metal foil. Now called the Rutherford gold foil experiment, or the Geiger–Marsden experiment, these measurements made the extraordinary discovery that although most alpha particles passing through a thin gold foil experienced little deflection, a few scattered to a high angle. The scattering indicated that some of the alpha particles ricocheted back from a small, but dense, component inside the atoms. Based on these measurements, by 1911 it was apparent to Rutherford that the atom consisted of a small, massive nucleus with positive charge surrounded by a much larger cloud of negatively charged electrons. The concentrated atomic mass was required to provide the observed deflection of the alpha particles, and Rutherford developed a mathematical model that accounted for the scattering.


title: Discovery of the neutron

text: The Rutherford model was very influential, motivating the Bohr model for electrons orbiting the nucleus in 1913 and eventually leading to quantum mechanics by the mid-1920s.


title: Discovery of the neutron

text: Concurrent with the work of Rutherford, Geiger, and Marsden, the radiochemist Frederick Soddy at the University of Glasgow was studying chemistry related problems on radioactive materials. Soddy had worked with Rutherford on radioactivity at McGill University. By 1910, about 40 different radioactive elements, referred to as "radioelements", had been identified between uranium and lead, although the periodic table only allowed for 11 elements. Soddy and Kazimierz Fajans independently found in 1913 that an element undergoing alpha decay will produce an element two places to the left in the periodic system and an element undergoing beta decay will produce an element one place to the right in the periodic system. Also, those radioelements that reside in the same places in the periodic system are chemically identical. Soddy called these chemically identical elements isotopes. For his study of radioactivity and the discovery of isotopes, Soddy was awarded the 1921 Nobel Prize in Chemistry.


title: Discovery of the neutron

text: Building from work by J. J. Thomson on the deflection of positively charged atoms by electric and magnetic fields, Francis Aston built the first mass spectrograph at the Cavendish Laboratory in 1919. His aim, which he easily achieved, was to separate the two isotopes of neon, and . Aston discovered that the masses of all the particles are whole numbers (the whole number rule): that is, the masses of all the isotopes are whole number multiples of the mass of the hydrogen atom. In these measurements, Aston arbitrarily computed his masses relative to oxygen-16, which he took to have a mass of exactly 16. (Today the atomic mass unit (amu) is relative to carbon-12.) Ironically, the one exception to this rule was hydrogen itself, which had a mass value of 1.008. The excess mass was small, but well outside the limits of experimental uncertainty. Aston and others quickly realized that the discrepancy is due to the binding energy of atoms, that is, the mass of a number of hydrogen atoms bound into a single atom must be less than the sum of the masses of the separate hydrogen atoms. Aston's work on isotopes won him the 1922 Nobel Prize in Chemistry for the discovery of isotopes in a large number of non-radioactive elements, and for his enunciation of the whole number rule.


title: Discovery of the neutron

text: Rutherford and others had noted the disparity between the mass of an atom, computed in atomic mass units, and the approximate charge required on the nucleus for the Rutherford model to work. The required charge of the atomic nucleus was usually about half its atomic mass.


title: Discovery of the neutron

text: Antonius van den Broek boldly hypothesized that the required charge, denoted by "Z", was not half of the atomic weight for elements, but instead was exactly equal to the element's ordinal position in the periodic table. At that time, the positions of the elements in the periodic table were not known to have any physical significance. If the elements were ordered based on increasing atomic mass, however, periodicity in chemical properties was exhibited. Exceptions to this periodicity were apparent, however, such as cobalt and nickel.


title: Discovery of the neutron

text: At the University of Manchester in 1913 Henry Moseley discussed the new Bohr model of the atom with the visiting Bohr. The model accounted for the electromagnetic emission spectrum from the hydrogen atom, and Moseley and Bohr wondered if the electromagnetic emission spectra of heavier elements such as cobalt and nickel would follow their ordering by weight, or by their position in the periodic table. In 1913–1914 Moseley tested the question experimentally by using X-ray diffraction techniques. He found that the most intense short-wavelength line in the X-ray spectrum of a particular element, known as the K-alpha line, was related to the element's position in the periodic table, that is, its atomic number, "Z". Indeed, Moseley introduced this nomenclature. Moseley found that the frequencies of the radiation were related in a simple way to the atomic number of the elements for a large number of elements.


title: Discovery of the neutron

text: Within a year it was noted that the equation for the relation, now called Moseley's law, could be explained in terms of the 1913 Bohr model, with reasonable extra assumptions about atomic structure in other elements. Moseley's result, by Bohr's later account, not only established atomic number as a measurable experimental quantity, but gave it a physical meaning as the positive charge on the atomic nucleus. The elements could be ordered in the periodic system in order of atomic number, rather than atomic weight. The result tied together the organization of the periodic table, the Bohr model for the atom, and Rutherford's model for alpha scattering from nuclei. It was cited by Rutherford, Bohr, and others as a critical advance in understanding the nature of the atomic nucleus.


title: Discovery of the neutron

text: Further research in atomic physics was interrupted by the outbreak of World War I. Moseley was killed in 1915 at the Battle of Gallipoli, while Rutherford's student James Chadwick was interned in Germany for the duration of the war, 1914–1918. In Berlin, Lise Meitner's and Otto Hahn's research work on determining the radioactive decay chains of radium and uranium by precise chemical separation was interrupted. Meitner spent much of the war working as a radiologist and medical X-ray technician near the Austrian front, while Hahn, a chemist, worked on research in poison gas warfare.


title: Discovery of the neutron

text: In 1920 Rutherford gave a Bakerian lecture at the Royal Society entitled the "Nuclear Constitution of Atoms", a summary of recent experiments on atomic nuclei and conclusions as to the structure of atomic nuclei. By 1920, the existence of electrons within the atomic nucleus was widely assumed. It was assumed the nucleus consisted of hydrogen nuclei in number equal to the atomic mass. But since each hydrogen nucleus had charge +1, the nucleus required a smaller number of "internal electrons" each of charge −1 to give the nucleus its correct total charge. The mass of protons is about 1800 times greater than that of electrons, so the mass of the electrons is incidental in this computation. Such a model was consistent with the scattering of alpha particles from heavy nuclei, as well as the charge and mass of the many isotopes that had been identified. There were other motivations for the proton–electron model. As noted by Rutherford at the time, "We have strong reason for believing that the nuclei of atoms contain electrons as well as positively charged bodies...", namely, it was known that beta radiation was electrons emitted from the nucleus.


title: Discovery of the neutron

text: In that lecture, Rutherford conjectured the existence of new particles. The alpha particle was known to be very stable, and it was assumed to retain its identity within the nucleus. The alpha particle was presumed to consist of four protons and two closely bound electrons to give it +2 charge and mass 4. In a 1919 paper, Rutherford had reported the apparent discovery of a new doubly charged particle of mass 3, denoted the X++, interpreted to consist of three protons and a closely bound electron. This result suggested to Rutherford the likely existence of two new particles: one of two protons with a closely bound electron, and another of one proton and a closely bound electron. The X++ particle was later determined to have mass 4 and to be just a low-energy alpha particle. Nevertheless, Rutherford had conjectured the existence of the deuteron, a +1 charge particle of mass 2, and the neutron, a neutral particle of mass 1. The former is the nucleus of deuterium, discovered in 1931 by Harold Urey. The mass of the hypothetical neutral particle would be little different from that of the proton. Rutherford determined that such a zero-charge particle would be difficult to detect by available techniques.


title: Discovery of the neutron

text: About the time of Rutherford's lecture, other publications appeared with similar suggestions of a proton–electron composite in the nucleus, and in 1921 William Harkins, an American chemist, named the uncharged particle the "neutron". About that same time the word "proton" was adopted for the hydrogen nucleus. Neutron was apparently constructed from the Latin root for "neutral" and the Greek ending "-on" (by imitation of "electron" and "proton"). References to the word "neutron" in connection with the atom can be found in the literature as early as 1899, however.


title: Discovery of the neutron

text: Rutherford and Chadwick immediately began an experimental program at the Cavendish Laboratory in Cambridge to search for the neutron. The experiments continued throughout the 1920s without success.


title: Discovery of the neutron

text: Rutherford's conjecture and the hypothetical "neutron" were not widely accepted. In his 1931 monograph on the "Constitution of Atomic Nuclei and Radioactivity", George Gamow, then at the Institute for Theoretical Physics in Copenhagen, did not mention the neutron. At the time of their 1932 measurements in Paris that would lead to the discovery of the neutron, Irène Joliot-Curie and Frédéric Joliot were unaware of the conjecture.


title: Discovery of the neutron

text: Throughout the 1920s, physicists assumed that the atomic nucleus was composed of protons and "nuclear electrons". Under this hypothesis, the nitrogen-14 (N) nucleus would be composed of 14 protons and 7 electrons, so that it would have a net charge of +7 elementary charge units and a mass of 14 atomic mass units. This nucleus would also be orbited by another 7 electrons, termed "external electrons" by Rutherford, to complete the N atom. However problems with the hypothesis soon became apparent.


title: Discovery of the neutron

text: Ralph Kronig pointed out in 1926 that the observed hyperfine structure of atomic spectra was inconsistent with the proton–electron hypothesis. This structure is caused by the influence of the nucleus on the dynamics of orbiting electrons. The magnetic moments of supposed "nuclear electrons" should produce hyperfine spectral line splittings similar to the Zeeman effect, but no such effects were observed. It seemed that the magnetic moment of the electron vanished when it was within the nucleus.


title: Discovery of the neutron

text: While on a visit to Utrecht University in 1928, Kronig learned of a surprising aspect of the rotational spectrum of N. The precision measurement made by Leonard Ornstein, the director of Utrecht's Physical Laboratory, showed that the spin of nitrogen nucleus must be equal to one. However, if the nitrogen-14 (N) nucleus was composed of 14 protons and 7 electrons, an odd number of spin-1/2 particles, then the resultant nuclear spin should be half-integer. Kronig therefore suggested that perhaps "protons and electrons do not retain their identity to the extent they do outside the nucleus".


title: Discovery of the neutron

text: Observations of the rotational energy levels of diatomic molecules using Raman spectroscopy by Franco Rasetti in 1929 were inconsistent with the statistics expected from the proton–electron hypothesis. Rasetti obtained band spectra for H and N molecules. While the lines for both diatomic molecules showed alternation in intensity between light and dark, the pattern of alternation for H is opposite to that of the N. After carefully analyzing these experimental results, German physicists Walter Heitler and Gerhard Herzberg showed that the hydrogen nuclei obeys Fermi statistics and the nitrogen nuclei obeys Bose statistics. However, a then unpublished result of Eugene Wigner showed that a composite system with an odd number of spin-1/2 particles must obey Fermi statistics; a system with an even number of spin-1/2 particle obeys Bose statistics. If the nitrogen nucleus had 21 particles, it should obey Fermi statistics, contrary to fact. Thus, Heitler and Herzberg concluded: "the electron in the nucleus ... loses its ability to determine the statistics of the nucleus."


title: Discovery of the neutron

text: The Klein paradox, discovered by Oskar Klein in 1928, presented further quantum mechanical objections to the notion of an electron confined within a nucleus. Derived from the Dirac equation, this clear and precise paradox suggested that an electron approaching a high potential barrier has a high probability of passing through the barrier by a pair creation process. Apparently, an electron could not be confined within a nucleus by any potential well. The meaning of this paradox was intensely debated at the time.


title: Discovery of the neutron

text: By about 1930 it was generally recognized that it was difficult to reconcile the proton–electron model for nuclei with the Heisenberg uncertainty relation of quantum mechanics. This relation, , implies that an electron confined to a region the size of an atomic nucleus typically has a kinetic energy of about 40 MeV, which is larger than the observed energy of beta particles emitted from the nucleus. Such energy is also much larger than the binding energy of nucleons, which Aston and others had shown to be less than 9 MeV per nucleon.


title: Discovery of the neutron

text: In 1927, Charles Ellis and W. Wooster at the Cavendish Laboratory measured the energies of β-decay electrons. They found that the distribution of energies from any particular radioactive nuclei was broad and continuous, a result that contrasted notably with the distinct energy values observed in alpha and gamma decay. Further, the continuous energy distribution seemed to indicate that energy was not conserved by this "nuclear electrons" process. Indeed, in 1929 Bohr proposed to modify the law of energy conservation to account for the continuous energy distribution. The proposal earned the support of Werner Heisenberg. Such considerations were apparently reasonable, inasmuch as the laws of quantum mechanics had so recently overturned the laws of classical mechanics.


title: Discovery of the neutron

text: While all these considerations did not "prove" an electron could not exist in the nucleus, they were confusing and challenging for physicists to interpret. Many theories were invented to explain how the above arguments could be wrong. In his 1931 monograph, Gamow summarized all these contradictions, marking the statements regarding electrons in the nucleus with warning symbols.


title: Discovery of the neutron

text: In 1930, Walther Bothe and Herbert Becker in Giessen, Germany found that if the energetic alpha particles emitted from polonium fell on certain light elements, specifically beryllium (), boron (), or lithium (), an unusually penetrating radiation was produced. Beryllium produced the most intense radiation. Polonium is highly radioactive, producing energetic alpha radiation, and it was commonly used for scattering experiments at the time. Alpha radiation can be influenced by an electric field, because it is composed of charged particles. The observed penetrating radiation was not influenced by an electric field, however, so it was thought to be gamma radiation. The radiation was more penetrating than any gamma rays known, and the details of experimental results were difficult to interpret.


title: Discovery of the neutron

text: Two years later Irène Joliot-Curie and Frédéric Joliot in Paris showed that if this unknown radiation fell on paraffin wax, or any other hydrogen-containing compound, it ejected protons of very high energy (5 MeV). This observation was not in itself inconsistent with the assumed gamma ray nature of the new radiation, but that interpretation (Compton scattering) had a logical problem. From energy and momentum considerations, a gamma ray would have to have impossibly high energy (50 MeV) to scatter a massive proton. In Rome, the young physicist Ettore Majorana declared that the manner in which the new radiation interacted with


title: Discovery of the neutron

text: On hearing of the Paris results, neither Rutherford nor James Chadwick at the Cavendish Laboratory believed the gamma ray hypothesis. Assisted by Norman Feather, Chadwick quickly performed a series of experiments showing that the gamma ray hypothesis was untenable. The previous year, Chadwick, J.E.R. Constable, and E.C. Pollard had already conducted experiments on disintegrating light elements using alpha radiation from polonium. They had also developed more accurate and efficient methods for detecting, counting, and recording the ejected protons. Chadwick repeated the creation of the radiation using beryllium to absorb the alpha particles: Following the Paris experiment, he aimed the radiation at paraffin wax, a hydrocarbon high in hydrogen content, hence offering a target dense with protons. As in the Paris experiment, the radiation energetically scattered some of the protons. Chadwick measured the range of these protons, and also measured how the new radiation impacted the atoms of various gases. He found that the new radiation consisted of not gamma rays, but uncharged particles with about the same mass as the proton. These particles were neutrons. Chadwick won the Nobel Prize in Physics in 1935 for this discovery.


title: Discovery of the neutron

text: The year 1932 was later referred to as the "annus mirabilis" for nuclear physics in the Cavendish Laboratory, with discoveries of the neutron, artificial nuclear disintegration by the Cockcroft–Walton particle accelerator, and the positron.


title: Discovery of the neutron

text: Given the problems of the "proton–electron model", it was quickly accepted that the atomic nucleus is composed of protons and neutrons, although the precise nature of the neutron was initially unclear. Within months after the discovery of the neutron, Werner Heisenberg and Dmitri Ivanenko had proposed proton–neutron models for the nucleus. Heisenberg's landmark papers approached the description of protons and neutrons in the nucleus through quantum mechanics. While Heisenberg's theory for protons and neutrons in the nucleus was a "major step toward understanding the nucleus as a quantum mechanical system," he still assumed the presence of nuclear electrons. In particular, Heisenberg assumed the neutron was a proton–electron composite, for which there is no quantum mechanical explanation. Heisenberg had no explanation for how lightweight electrons could be bound within the nucleus. Heisenberg introduced the first theory of nuclear exchange forces that bind the nucleons. He considered protons and neutrons to be different quantum states of the same particle, i.e., nucleons distinguished by the value of their nuclear isospin quantum numbers.


title: Discovery of the neutron

text: The proton–neutron model explained the puzzle of dinitrogen. When N was proposed to consist of 3 pairs each of protons and neutrons, with an additional unpaired neutron and proton each contributing a spin of  ħ in the same direction for a total spin of 1 ħ, the model became viable. Soon, neutrons were used to naturally explain spin differences in many different nuclides in the same way.


title: Discovery of the neutron

text: If the proton–neutron model for the nucleus resolved many issues, it highlighted the problem of explaining the origins of beta radiation. No existing theory could account for how electrons, or positrons, could emanate from the nucleus. In 1934, Enrico Fermi published his classic paper describing the process of beta decay, in which the neutron decays to a proton by "creating" an electron and a (as yet undiscovered) neutrino. The paper employed the analogy that photons, or electromagnetic radiation, were similarly created and destroyed in atomic processes. Ivanenko had suggested a similar analogy in 1932. Fermi's theory requires the neutron to be a spin- particle. The theory preserved the principle of conservation of energy, which had been thrown into question by the continuous energy distribution of beta particles. The basic theory for beta decay proposed by Fermi was the first to show how particles could be created and destroyed. It established a general, basic theory for the interaction of particles by weak or strong forces. While this influential paper has stood the test of time, the ideas within it were so new that when it was first submitted to the journal Nature in 1933 it was rejected as being too speculative.


title: Discovery of the neutron

text: The question of whether the neutron was a composite particle of a proton and an electron persisted for a few years after its discovery. In 1932 Harrie Massey explored a model for a composite neutron to account for its great penetrating power through matter and its electrical neutrality, for example. The issue was a legacy of the prevailing view from the 1920s that the only elementary particles were the proton and electron.


title: Discovery of the neutron

text: The nature of the neutron was a primary topic of discussion at the 7th Solvay Conference held in October 1933, attended by Heisenberg, Niels Bohr, Lise Meitner, Ernest Lawrence, Fermi, Chadwick, and others. As posed by Chadwick in his Bakerian Lecture in 1933, the primary question was the mass of the neutron relative to the proton. If the neutron's mass was less than the combined masses of a proton and an electron (), then the neutron could be a proton-electron composite because of the mass defect from the nuclear binding energy. If greater than the combined masses, then the neutron was elementary like the proton. The question was challenging to answer because the electron's mass is only 0.05% of the proton's, hence exceptionally precise measurements were required.


title: Discovery of the neutron

text: The difficulty of making the measurement is illustrated by the wide-ranging values for the mass of the neutron obtained from 1932 to 1934. The accepted value today is . In Chadwick's 1932 paper reporting on the discovery, he estimated the mass of the neutron to be between and . By bombarding boron with alpha particles, Frédéric and Irène Joliot-Curie obtained a high value of , while Ernest Lawrence's team at the University of California measured the small value using their new cyclotron.


title: Discovery of the neutron

text: In 1935 Chadwick and his doctoral student Maurice Goldhaber resolved the issue by reporting the first accurate measurement of the mass of the neutron. They used the 2.6 MeV gamma rays of Thallium-208 (Tl) (then known as thorium C") to photodisintegrate the deuteron


title: Discovery of the neutron

text: In this reaction, the resulting proton and neutron have about equal kinetic energy, since their masses are about equal. The kinetic energy of the resulting proton could be measured (0.24 MeV), and therefore the deuteron's binding energy could be determined (2.6 MeV − 2(0.24 MeV) = 2.1 MeV, or ). The neutron's mass could then be determined by the simple mass balance


title: Discovery of the neutron

text: where m refer to the deuteron, proton, or neutron mass, and "b.e." is the binding energy. The masses of the deuteron and proton were known; Chadwick and Goldhaber used values 2.0142 u and 1.0081 u, respectively. They found that the neutron's mass was slightly greater than the mass of the proton or , depending on the precise value used for the deuteron mass. The mass of the neutron was too large to be a proton–electron composite, and the neutron was therefore identified as an elementary particle. Chadwick and Goldhaber predicted that a free neutron would be able to decay into a proton, electron, and neutrino (beta decay).


title: Discovery of the neutron

text: Soon after the discovery of the neutron, indirect evidence suggested the neutron had an unexpected non-zero value for its magnetic moment. Attempts to measure the neutron's magnetic moment originated with the discovery by Otto Stern in 1933 in Hamburg that the proton had an anomalously large magnetic moment. By 1934 groups led by Stern, now in Pittsburgh, and I. I. Rabi in New York had independently deduced that the magnetic moment of the neutron was negative and unexpectedly large by measuring the magnetic moments of the proton and deuteron. Values for the magnetic moment of the neutron were also determined by Robert Bacher (1933) at Ann Arbor and I.Y. Tamm and S.A. Altshuler (1934) in the Soviet Union from studies of the hyperfine structure of atomic spectra. By the late 1930s accurate values for the magnetic moment of the neutron had been deduced by the Rabi group using measurements employing newly developed nuclear magnetic resonance techniques. The large value for the proton's magnetic moment and the inferred negative value for the neutron's magnetic moment were unexpected and raised many questions.


title: Discovery of the neutron

text: The discovery of the neutron immediately gave scientists a new tool for probing the properties of atomic nuclei. Alpha particles had been used over the previous decades in scattering experiments, but such particles, which are helium nuclei, have +2 charge. This charge makes it difficult for alpha particles to overcome the Coulomb repulsive force and interact directly with the nuclei of atoms. Since neutrons have no electric charge, they do not have to overcome this force to interact with nuclei. Almost coincident with their discovery, neutrons were used by Norman Feather, Chadwick's colleague and protege, in scattering experiments with nitrogen. Feather was able to show that neutrons interacting with nitrogen nuclei scattered to protons or induced nitrogen to disintegrate to form boron with the emission of an alpha particle. Feather was therefore the first to show that neutrons produce nuclear disintegrations.


title: Discovery of the neutron

text: In Rome, Enrico Fermi and his team bombarded heavier elements with neutrons and found the products to be radioactive. By 1934 they had used neutrons to induce radioactivity in 22 different elements, many of these elements of high atomic number. Noticing that other experiments with neutrons at his laboratory seemed to work better on a wooden table than a marble table, Fermi suspected that the protons of the wood were slowing the neutrons and so increasing the chance for the neutron to interact with nuclei. Fermi therefore passed neutrons through paraffin wax to slow them and found that the radioactivity of some bombarded elements increased by a factor of tens to hundreds. The cross section for interaction with nuclei is much larger for slow neutrons than for fast neutrons. In 1938 Fermi received the Nobel Prize in Physics ""for his demonstrations of the existence of new radioactive elements produced by neutron irradiation, and for his related discovery of nuclear reactions brought about by slow neutrons"".


title: Discovery of the neutron

text: In Berlin, the collaboration of Lise Meitner and Otto Hahn, together with their assistant Fritz Strassmann, furthered the research begun by Fermi and his team when they bombarded uranium with neutrons. Between 1934 and 1938, Hahn, Meitner, and Strassmann found a great number of radioactive transmutation products from these experiments, all of which they regarded as transuranic. Transuranic nuclides are those that have an atomic number greater than uranium (92), formed by neutron absorption; such nuclides are not naturally occurring. In July 1938, Meitner was forced to escape antisemitic persecution in Nazi Germany after the Anschluss, and she was able to secure a new position in Sweden. The decisive experiment on 16–17 December 1938 (using a chemical process called "radium–barium–mesothorium fractionation") produced puzzling results: what they had understood to be three isotopes of radium were instead consistently behaving as barium. Radium (atomic number 88) and barium (atomic number 56) are in the same chemical group. By January 1939 Hahn had concluded that what they had thought were transuranic nuclides were instead much lighter nuclides, such as barium, lanthanum, cerium and light platinoids. Meitner and her nephew Otto Frisch immediately and correctly interpreted these observations as resulting from nuclear fission, a term coined by Frisch.


title: Discovery of the neutron

text: Hahn and his collaborators had detected the splitting of uranium nuclei, made unstable by neutron absorption, into lighter elements. Meitner and Frisch also showed that the fission of each uranium atom would release about 200 MeV of energy. The discovery of fission electrified the global community of atomic physicists and the public. In their second publication on nuclear fission, Hahn and Strassmann predicted the existence and liberation of additional neutrons during the fission process. Frédéric Joliot and his team proved this phenomenon to be a chain reaction in March 1939. In 1945 Hahn received the 1944 Nobel Prize in Chemistry ""for his discovery of the fission of heavy atomic nuclei.""


title: Discovery of the neutron

text: The discovery of nuclear fission at the end of 1938 marked a shift in the centers of nuclear research from Europe to the United States. Large numbers of scientists were migrating to the United States to escape the troubles and antisemitism in Europe and the looming war (See Jewish scientists and the Manhattan Project). The new centers of nuclear research were the universities in the United States, particularly Columbia University in New York and the University of Chicago where Enrico Fermi had relocated, and a secret research facility at Los Alamos, New Mexico, established in 1942, the new home of the Manhattan project. This wartime project was focussed on the construction of nuclear weapons, exploiting the enormous energy released by the fission of uranium or plutonium through neutron-based chain reactions.


title: Discovery of the neutron

text: The discoveries of the neutron and positron in 1932 were the start of the discoveries of many new particles. Muons were discovered in 1936. Pions and kaons were discovered in 1947, while lambda particles were discovered in 1950. Throughout the 1950s and 1960s, a large number of particles called hadrons were discovered. A classification scheme for organizing all these particles, proposed independently by Murray Gell-Mann and


title: Discovery of the neutron

text: George Zweig in 1964, became known as the quark model. By this model, particles such as the proton and neutron were not elementary, but composed of various configurations of a small number of other truly elementary particles called partons or quarks. The quark model received experimental verification beginning in the late 1960s and finally provided an explanation for the neutron's anomalous magnetic moment.


title: Skee-Lo

text: Antoine Roundtree (born March 27, 1975), better known by his stage name Skee-Lo, is an American rapper. He is best known for his 1995 song "I Wish", which became a hit in several countries and reached No. 13 on the "Billboard" Hot 100.


title: Skee-Lo

text: His first single "I Wish" became a hit on radio and TV during the summer of 1995 with a number one MTV video directed by Marty Thomas that won many nominations and wins from VMA Awards worldwide. The song was known for its lack of explicit lyrics at a time when gangsta rap was very popular. His debut album of the same title was released shortly thereafter, earning him two Grammy nominations for both the album and single. Afterwards he stopped his rapping career before coming back to release a new album in 2000 and another in 2012.


title: Skee-Lo

text: Born in Chicago, Illinois, Roundtree and his family subsequently relocated to and then moved to Riverside, California, when he was nine. He first started rapping in 1983.


title: Skee-Lo

text: He graduated from high school in Moreno Valley before moving back to Los Angeles in 1993. He attended El Camino College. At about this time, he converted to Islam and joined the Nation of Islam.


title: Skee-Lo

text: Skee-Lo's debut album "I Wish" was released in 1995, which he first started working on in 1993. The album was a huge success. One of the singles, "Top of the Stairs", was featured in the ending credits as well as the soundtrack to the 1995 film "Money Train". He quietly "retired" from rapping five months after the album's release because of a dispute with label Sunshine Records. Skee-Lo claims Sunshine Records took all the credit from the album's success, and he did not make a single cent from it. "So I refused to shoot any more videos, promote or record music. I'm not their slave. I wasn't working for free." After many years, he eventually won the rights against the label in court.


title: Skee-Lo

text: Skee-Lo recorded a cover of the Schoolhouse Rock! song "The Tale of Mr. Morton", which taught sentence structure (subject, verb and predicate). The song appears on the 1996 compilation album, "Schoolhouse Rock! Rocks".


title: Skee-Lo

text: He co-wrote "I'll Be Your Everything" by boy band Youngstown. The song is featured on the 1999 "Inspector Gadget" movie soundtrack.


title: Skee-Lo

text: He returned to the music scene in 2000 with a new single "I Can't Stop", which would be the titular track on his second album released in 2001.


title: Skee-Lo

text: In January 2010, Skee-Lo returned from another musical hiatus with the EP "Overdose" where he collaborated with record producer Michael DeBarge.


title: Skee-Lo

text: In 2011 he wrote "Burnin' Up" as part of the soundtrack of the 2013 horror movie "Killer Holiday" from Lionsgate Entertainment. MTV produced a music video directed by Marty Thomas that was filmed at actor Howie Mandel's house, which was rented by his son to MTV without Mandel's knowledge. It was released under his own indie label, Skeelo Musik.


title: Skee-Lo

text: Skee-Lo made a guest appearance on a track "Now You See My Life" with former Hollywood Undead rapper Deuce from Deuce's album "Nine Lives" in 2012.


title: Skee-Lo

text: He released his third album, "Fresh Ideas", on his own label Skeelo Musik, via iTunes on November 13, 2012. It appeared in stores July 9, 2013.


title: Airy disk

text: In optics, the Airy disk (or Airy disc) and Airy pattern are descriptions of the best-focused spot of light that a perfect lens with a circular aperture can make, limited by the diffraction of light. The Airy disk is of importance in physics, optics, and astronomy.


title: Airy disk

text: The diffraction pattern resulting from a uniformly illuminated, circular aperture has a bright central region, known as the Airy disk, which together with the series of concentric rings around is called the Airy pattern. Both are named after George Biddell Airy. The disk and rings phenomenon had been known prior to Airy; John Herschel described the appearance of a bright star seen through a telescope under high magnification for an 1828 article on light for the "Encyclopedia Metropolitana":


title: Airy disk

text: Airy wrote the first full theoretical treatment explaining the phenomenon (his 1835 "On the Diffraction of an Object-glass with Circular Aperture").


title: Airy disk

text: Mathematically, the diffraction pattern is characterized by the wavelength of light illuminating the circular aperture, and the aperture's size. The "appearance" of the diffraction pattern is additionally characterized by the sensitivity of the eye or other detector used to observe the pattern.


title: Airy disk

text: The most important application of this concept is in cameras, microscopes and telescopes. Due to diffraction, the smallest point to which a lens or mirror can focus a beam of light is the size of the Airy disk. Even if one were able to make a perfect lens, there is still a limit to the resolution of an image created by such a lens. An optical system in which the resolution is no longer limited by imperfections in the lenses but only by diffraction is said to be diffraction limited.


title: Airy disk

text: Far from the aperture, the angle at which the first minimum occurs, measured from the direction of incoming light, is given by the approximate formula:


title: Airy disk

text: where "θ" is in radians, "λ" is the wavelength of the light in meters, and "d" is the diameter of the aperture in meters. The full width at half maximum is given by formula_3.


title: Airy disk

text: where "s" was the angle of first minimum in seconds of arc, "a" was the radius of the aperture in inches, and the wavelength of light was assumed to be 0.000022 inches (560 nm; the mean of visible wavelengths). This is equal to the angular resolution of a circular aperture. The Rayleigh criterion for barely resolving two objects that are point sources of light, such as stars seen through a telescope, is that the center of the Airy disk for the first object occurs at the first minimum of the Airy disk of the second. This means that the angular resolution of a diffraction-limited system is given by the same formulae.


title: Airy disk

text: However, while the angle at which the first minimum occurs (which is sometimes described as the radius of the Airy disk) depends only on wavelength and aperture size, the appearance of the diffraction pattern will vary with the intensity (brightness) of the light source. Because any detector (eye, film, digital) used to observe the diffraction pattern can have an intensity threshold for detection, the full diffraction pattern may not be apparent. In astronomy, the outer rings are frequently not apparent even in a highly magnified image of a star. It may be that none of the rings are apparent, in which case the star image appears as a disk (central maximum only) rather than as a full diffraction pattern. Furthermore, fainter stars will appear as smaller disks than brighter stars, because less of their central maximum reaches the threshold of detection. While in theory all stars or other "point sources" of a given wavelength and seen through a given aperture have the same Airy disk radius characterized by the above equation (and the same diffraction pattern size), differing only in intensity, the appearance is that fainter sources appear as smaller disks, and brighter sources appear as larger disks. This was described by Airy in his original work:


title: Airy disk

text: The rapid decrease of light in the successive rings will sufficiently explain the visibility of two or three rings with a very bright star and the non-visibility of rings with a faint star. The difference of the diameters of the central spots (or spurious disks) of different stars ... is also fully explained. Thus the radius of the spurious disk of a faint star, where light of less than half the intensity of the central light makes no impression on the eye, is determined by ["s" = 1.17/"a"], whereas the radius of the spurious disk of a bright star, where light of 1/10 the intensity of the central light is sensible, is determined by ["s" = 1.97/"a"].


title: Airy disk

text: Despite this feature of Airy's work, the radius of the Airy disk is often given as being simply the angle of first minimum, even in standard textbooks. In reality, the angle of first minimum is a limiting value for the size of the Airy disk, and not a definite radius.


title: Airy disk

text: If two objects imaged by a camera are separated by an angle small enough that their Airy disks on the camera detector start overlapping, the objects cannot be clearly separated any more in the image, and they start blurring together. Two objects are said to be "just resolved" when the maximum of the first Airy pattern falls on top of the first minimum of the second Airy pattern (the Rayleigh criterion).


title: Airy disk

text: Therefore, the smallest angular separation two objects can have before they significantly blur together is given as stated above by


title: Airy disk

text: Thus, the ability of the system to resolve detail is limited by the ratio of λ/"d". The larger the aperture for a given wavelength, the finer the detail that can be distinguished in the image.


title: Airy disk

text: where formula_7 is the separation of the images of the two objects on the film, and formula_8 is the distance from the lens to the film.


title: Airy disk

text: If we take the distance from the lens to the film to be approximately equal to the focal length of the lens, we find


title: Airy disk

text: but formula_10 is the f-number of a lens. A typical setting for use on an overcast day would be (see Sunny 16 rule). For violet, the shortest wavelength visible light, the wavelength λ is about 420 nanometers (see cone cells for sensitivity of S cone cells). This gives a value for formula_7 of about 4 µm. In a digital camera, making the pixels of the image sensor smaller than half this value (one pixel for each object, one for each space between) would not significantly increase the captured image resolution. However, it may improve the final image by over-sampling, allowing noise reduction.


title: Airy disk

text: The fastest f-number for the human eye is about 2.1, corresponding to a diffraction-limited point spread function with approximately 1 μm diameter. However, at this f-number, spherical aberration limits visual acuity, while a 3 mm pupil diameter (f/5.7) approximates the resolution achieved by the human eye. The maximum density of cones in the human fovea is approximately 170,000 per square millimeter, which implies that the cone spacing in the human eye is about 2.5 μm, approximately the diameter of the point spread function at f/5.


title: Airy disk

text: A circular laser beam with uniform intensity across the circle (a flat-top beam) focused by a lens will form an Airy disk pattern at the focus. The size of the Airy disk determines the laser intensity at the focus.


title: Airy disk

text: Some weapon aiming sights (e.g. FN FNC) require the user to align a peep sight (rear, nearby sight, i.e. which will be out of focus) with a tip (which should be focused and overlaid on the target) at the end of the barrel. When looking through the peep sight, the user will notice an Airy disk that will help center the sight over the pin.


title: Airy disk

text: Light from a uniformly illuminated circular aperture (or from a uniform, flattop beam) will exhibit an Airy diffraction pattern far away from the aperture due to Fraunhofer diffraction (far-field diffraction).


title: Airy disk

text: The conditions for being in the far field and exhibiting an Airy pattern are: the incoming light illuminating the aperture is a plane wave (no phase variation across the aperture), the intensity is constant over the area of the aperture, and the distance formula_12 from the aperture where the diffracted light is observed (the screen distance) is large compared to the aperture size, and the radius formula_13 of the aperture is not too much larger than the wavelength formula_14 of the light. The last two conditions can be formally written as formula_15 .


title: Airy disk

text: In practice, the conditions for uniform illumination can be met by placing the source of the illumination far from the aperture. If the conditions for far field are not met (for example if the aperture is large), the far-field Airy diffraction pattern can also be obtained on a screen much closer to the aperture by using a lens right after the aperture (or the lens itself can form the aperture). The Airy pattern will then be formed at the focus of the lens rather than at infinity.


title: Airy disk

text: Hence, the focal spot of a uniform circular laser beam (a flattop beam) focused by a lens will also be an Airy pattern.


title: Airy disk

text: In a camera or imaging system an object far away gets imaged onto the film or detector plane by the objective lens, and the far field diffraction pattern is observed at the detector. The resulting image is a convolution of the ideal image with the Airy diffraction pattern due to diffraction from the iris aperture or due to the finite size of the lens. This leads to the finite resolution of a lens system described above.


title: Airy disk

text: The intensity of the Airy pattern follows the Fraunhofer diffraction pattern of a circular aperture, given by the squared modulus of the Fourier transform of the circular aperture:


title: Airy disk

text: where formula_17 is the maximum intensity of the pattern at the Airy disc center, formula_18 is the Bessel function of the first kind of order one, formula_19 is the wavenumber, formula_13 is the radius of the aperture, and formula_21 is the angle of observation, i.e. the angle between the axis of the circular aperture and the line between aperture center and observation point. formula_22, where "q" is the radial distance from the observation point to the optical axis and "R" is its distance to the aperture. Note that the Airy disk as given by the above expression is only valid for large "R", where Fraunhofer diffraction applies; calculation of the shadow in the near-field must rather be handled using Fresnel diffraction.


title: Airy disk

text: However the exact Airy pattern "does" appear at a finite distance if a lens is placed at the aperture. Then the Airy pattern will be perfectly focussed at the distance given by the lens's focal length (assuming collimated light incident on the aperture) given by the above equations.


title: Airy disk

text: The zeros of formula_23 are at formula_24. From this, it follows that the first dark ring in the diffraction pattern occurs where formula_25, or


title: Airy disk

text: If a lens is used to focus the Airy pattern at a finite distance, then the radius formula_27 of the first dark ring on the focal plane is solely given by the numerical aperture "A" (closely related to the f-number) by


title: Airy disk

text: where the numerical aperture "A" is equal to the aperture's radius "d"/2 divided by R', the distance from the center of the Airy pattern to the "edge" of the aperture. Viewing the aperture of radius "d"/2 and lens as a camera (see diagram above) projecting an image onto a focal plane at distance "f", the numerical aperture "A" is related to the commonly-cited f-number "N= f/d" (ratio of the focal length to the lens diameter) according to formula_29; for "N"≫1 it is simply approximated as formula_30. This shows that the best possible image resolution of a camera is limited by the numerical aperture (and thus f-number) of its lens due to diffraction.


title: Airy disk

text: The half maximum of the central Airy disk (where formula_31) occurs at formula_32; the 1/e point (where formula_33) occurs at formula_34, and the maximum of the first ring occurs at formula_35.


title: Airy disk

text: The intensity formula_17 at the center of the diffraction pattern is related to the total power formula_37 incident on the aperture by


title: Airy disk

text: where formula_39 is the source strength per unit area at the aperture, A is the area of the aperture (formula_40) and R is the distance from the aperture. At the focal plane of a lens, formula_41. The intensity at the maximum of the first ring is about 1.75% of the intensity at the center of the Airy disk.


title: Airy disk

text: The expression for formula_42 above can be integrated to give the total power contained in the diffraction pattern within a circle of given size:


title: Airy disk

text: where formula_44 and formula_18 are Bessel functions. Hence the fractions of the total power contained within the first, second, and third dark rings (where formula_46) are 83.8%, 91.0%, and 93.8% respectively.


title: Airy disk

text: The Airy pattern falls rather slowly to zero with increasing distance from the center, with the outer rings containing a significant portion of the integrated intensity of the pattern. As a result, the root mean square (RMS) spotsize is undefined (i.e. infinite). An alternative measure of the spot size is to ignore the relatively small outer rings of the Airy pattern and to approximate the central lobe with a Gaussian profile, such that


title: Airy disk

text: where formula_48 is the irradiance at the center of the pattern, formula_49 represents the radial distance from the center of the pattern, and formula_50 is the Gaussian RMS width (in one dimension). If we equate the peak amplitude of the Airy pattern and Gaussian profile, that is, formula_51, and find the value of formula_50 giving the optimal approximation to the pattern, we obtain


title: Airy disk

text: where "N" is the f-number. If, on the other hand, we wish to enforce that the Gaussian profile has the same volume as does the Airy pattern, then this becomes


title: Airy disk

text: In optical aberration theory, it is common to describe an imaging system as "diffraction-limited" if the Airy disk radius is larger than the RMS spotsize determined from geometric ray tracing (see Optical lens design). The Gaussian profile approximation provides an alternative means of comparison: using the approximation above shows that the Gaussian waist formula_50 of the Gaussian approximation to the Airy disk is about two-third the Airy disk radius, i.e. formula_56 as opposed to formula_57.


title: Airy disk

text: Similar equations can also be derived for the obscured Airy diffraction pattern which is the diffraction pattern from an annular aperture or beam, i.e. a uniform circular aperture (beam) obscured by a circular block at the center. This situation is relevant to many common reflector telescope designs that incorporate a secondary mirror, including Newtonian telescopes and Schmidt–Cassegrain telescopes.


title: Airy disk

text: where formula_59 is the annular aperture obscuration ratio, or the ratio of the diameter of the obscuring disk and the diameter of the aperture (beam). formula_60, and x is defined as above: formula_61 where formula_12 is the radial distance in the focal plane from the optical axis, formula_14 is the wavelength and formula_64 is the f-number of the system. The fractional encircled energy (the fraction of the total energy contained within a circle of radius formula_12 centered at the optical axis in the focal plane) is then given by:


title: Airy disk

text: The practical effect of having a central obstruction in a telescope is that the central disc becomes slightly smaller, and the first bright ring becomes brighter at the expense of the central disc. This becomes more problematic with short focal length telescopes which require larger secondary mirrors.


title: Airy disk

text: A circular laser beam with uniform intensity profile, focused by a lens, will form an Airy pattern at the focal plane of the lens. The intensity at the center of the focus will be formula_68 where formula_37 is the total power of the beam, formula_70 is the area of the beam (formula_71 is the beam diameter), formula_14 is the wavelength, and formula_8 is the focal length of the lens.


title: Airy disk

text: A Gaussian beam transmitted through a hard aperture will be clipped. Energy is lost and edge diffraction occurs effectively increasing the divergence. Because of these effects there is a Gaussian beam diameter which maximizes the intensity in the far field. This occurs when the formula_74 diameter of the Gaussian is 89% of the aperture diameter, and the on axis intensity in the far field will be 81% of that produced by a uniform intensity profile.


title: Tiran Porter

text: Tiran Calvin Porter (born September 26, 1948) is an American bass and guitar player, vocalist and composer, best known as a member of The Doobie Brothers from 1972 to 1980 and 1987 to 1992.


title: Tiran Porter

text: Porter graduated from Leuzinger High School in Lawndale, California in 1966. He rose to fame as a member of the Doobie Brothers, replacing bassist Dave Shogren on their second album "Toulouse Street" in 1972.


title: Tiran Porter

text: His vocals were mostly restricted to the background in the studio, although he wrote and sang "For Someone Special" (a tribute to ill bandleader Tom Johnston) on the album "Takin' It To The Streets" (1976) and the creatively syncopated "Need A Lady" on the album "Livin' On The Fault Line" (1977). In concert, Porter usually performed lead vocals on one or two songs.


title: Tiran Porter

text: Porter's most notable contributions to the Doobie Brothers' sound were his busy and punchy bass lines; his distinctive tone permeates all of the band's classic compositions and hits. Often pictured with his trusty Alembic or Gibson basses during the seventies (usually with a Thunderbird or Ripper bass and earlier with an EB-0L), Porter played Fender instruments, most notably the Fender Jazz Bass, along with BC Rich Eagles and Rickenbackers.


title: Tiran Porter

text: His early technique, with rich chordal attack, was based mostly on his picking style, favoring the guitar pick over fingerstyle playing. The hit title track from "Takin' It to the Streets", which prominently features Porter's thundering, picked notes, is a prime example of this technique. More recently, even performing the old repertoire with the Doobies, Porter has been playing new, custom-made instruments almost exclusively finger style.


title: Tiran Porter

text: Porter left the Doobies in 1980, citing frustration with the hectic and constant touring schedule. His replacement was session man Willie Weeks, later famous for his collaboration with Michael Jackson and other Quincy Jones protégés. After guesting onstage with his former bandmates briefly during the 1982 farewell tour, he rejoined the Doobies in 1987. Porter played on "Cycles" (1989), whose title was taken from an unused song he wrote, and "Brotherhood" (1991). Neither album featured a Porter composition or lead vocal, and his bass is often buried in the mix. After five years of touring in support of "Cycles" and "Brotherhood", Porter finally quit the Doobies for good in 1992. He was reportedly still frustrated with constant touring and the band's preference for recording familiar sounding material instead of his own, more diverse compositions.


title: Tiran Porter

text: Porter released a self-produced solo album, "Playing to an Empty House", in 1995. It is a mix of rock, progressive, and jazz spotlighting Porter on all of the instruments and vocals. The album features jazz solos and little or no bass guitar, focusing instead on lead guitar and sequenced keyboards.


title: Tiran Porter

text: More recently, Porter played bass with singer-songwriter Keith Greeninger from Santa Cruz, California, Beatles tribute band White Album Ensemble, Stormin' Norman and the Cyclones, and Moby Grape during its occasional reunions.


title: The Velvet Underground discography

text: The discography of the American rock band the Velvet Underground consists of five studio albums, six live albums, 14 compilation albums, six box sets and eleven singles.


title: The Velvet Underground discography

text: The first line-up was formed in New York City consisting of Lou Reed on vocals and guitar, John Cale on several instruments (viola, keyboards and bass), Sterling Morrison on guitar and bass and Angus MacLise on percussion (replaced by Maureen Tucker in November 1965). On March 12, 1967 they released their debut album "The Velvet Underground & Nico" featuring German singer Nico. The album charted in the United States and originally peaked at number 171 on the "Billboard" album charts and produced two singles, "All Tomorrow's Parties" and "Sunday Morning", which did not chart anywhere. The album, produced by artist Andy Warhol, recharted in 2013 peaking at number 129.


title: The Velvet Underground discography

text: VU released their second studio album "White Light/White Heat", which peaked at number 199 of the "Billboard" charts. The album was more experimental than their first album, featuring a loud and aggressive musical style. The album was produced without Warhol and Nico for the first time.<ref name="White Light/White Heat"></ref> In March 1969 they released the third studio album "The Velvet Underground", their first project with Cale's replacement Doug Yule. It did not chart upon its original release. However, when reissued in 1985 it peaked at number 197 in the "Billboard" chart. One year later, "Loaded" was released by Atlantic Records, the first time by a major label. Compared with VU first releases, "Loaded" was produced for the mainstream, without thematizing sex and drugs. Although the songs were successful and popular in several music radios, neither singles, nor the album itself peaked in any music chart. It was the last album featuring Lou Reed, the last remaining founder of VU. After his departure, Yule became the new frontman of The Velvet Underground and toured together with Willie Alexander (keyboard), Walter Powers (bass guitar), Maureen Tucker (drums) and other musicians. In 1973, he recorded their fifth and last regular studio album, "Squeeze", which for some time dropped out of the official discography. After their break-off, several compilation albums were released, some of which feature outtakes from their previous studio sessions, most notably "VU", which peaked at number 85 on "Billboard" and 47 on the UK Albums Chart.


title: Noakhali riots

text: The Noakhali riots were a series of semi-organized massacres, rapes and abductions, combined with looting and arson of Hindu properties, perpetrated by the Muslim community in the districts of Noakhali in the Chittagong Division of Bengal (now in Bangladesh) in October–November 1946, a year before India's independence from British rule.