title: Eighth Doctor

text: Eventually, after a woman Sabbath loved sacrificed herself to save the Doctor from a malfunctioning time machine, Sabbath tore out the Doctor's second heart, allowing the Doctor to begin growing a new one.


title: Eighth Doctor

text: Shortly after the restoration of his heart, the Doctor found himself locked in a desperate struggle with Sabbath as, along with his mysterious business associates, Sabbath hatched a plan to destroy all alternate realities. Sabbath believed that time travellers like the Doctor, every time they landed somewhere, created an alternate reality where they didn't show up, and that the universe was unable to support so many alternates without suffering damage; therefore, he attempted to trigger an explosion at Event One – the Big Bang – that would erase all alternate universes and leaving only one possible timeline. However, Sabbath's allies had been lying to him; in reality, Time would only split if absolutely necessary, and even then, it was nearly impossible to travel between alternate realities. In reality, Sabbath had been manipulated into creating a plan that would effectively wipe out free will itself, with the Doctor proving that Sabbath's perception was wrong when they found themselves in a situation that could only have occurred due to time travel in a single timeline.


title: Eighth Doctor

text: The explosion at Event One was averted when the Doctor diverted the black light that would have triggered the explosion to 1898, but instead of reality collapsing into a single controllable timeline, what occurred was reality starting to 'slide' between histories, multiple parallel realities fighting to become the dominant one. Along with new companion Trix, the Doctor, Fitz and Anji travelled through the realities, including a world where the computer was never invented, a world where Earth was devastated by a strange even as all life aged forty years in 1843, and a world where humanity had time-travelling tours, with the Doctor being forced to erase at least two of these realities as he sought to restore the original timeline. During this adventure, the Doctor appeared to become a bit more cold and calculating, sacrificing an innocent man to escape a pocket universe and even leaving alternate versions of Fitz and Anji to die to preserve continuity. However, in the end, their sacrifices paid off, the Doctor managing to stabilise reality by resolving a paradox that had been hanging over them since the beginning of the crisis, and then, with Sabbath's help, they confronted his masters; the Council of Eight, mysterious beings who gained power by foreseeing likely future events and then ensuring that they came to pass. The Doctor, as a rogue element existing outside of Time, was the only unpredictable factor in their universe, and was thus the only person who could stop them. Ironically, it was Sabbath himself who gave the Doctor the edge needed to stop the council; realising that one of the Council members expected Sabbath to shoot the Council member with a weapon designed to send the subject into the Time Vortex, Sabbath instead shot himself, condemning himself to eternal agony in the Time Vortex, but completely undoing the council's plan and destroying their space station, simultaneously restoring the possibility of parallel worlds to the universe.


title: Eighth Doctor

text: Some while after this, the Doctor was captured by Marnal, one of the few surviving Gallifreyans, and accused of destroying Gallifrey. Although Gallifrey had been all but wiped from history by the Doctor's actions, Marnal was able to jury-rig a Time Space Visualiser to witness the Doctor actually push the button as he faced off against the Grandfather, although there were about three minutes between the Doctor firing the shot and Gallifrey's actual destruction where the Doctor's activities in the TARDIS were unaccounted for. Reflecting on his discoveries in the TARDIS, the Doctor, along with the aid of K9 (who had been transported into a hidden area of the TARDIS and trapped there following Gallifrey's destruction) realised that his memory loss hadn't been caused by the trauma of destroying Gallifrey; in fact, the Doctor had wiped his memories to give his mind space to store the contents of the Matrix within his brain, compressed down so he wouldn't be driven mad by the voices of all the dead Time Lords within him, his own memories presumably stored somewhere else in his mind, given his occasional flashes from his past. However, right then, the Doctor had more immediate worries; namely, saving Earth from a species of massive fly-like aliens called the Vore, who would soon have the power to devour the planet. As "The Gallifrey Chronicles" ended, the Doctor, Fitz, and Trix dove into the Vore mountain, the Doctor equipped with a plan to stop the Vore and save the world.


title: Eighth Doctor

text: Paul McGann first returned to his role as the Eighth Doctor in "Storm Warning", the first in a series of audio plays. During his time with Big Finish, McGann played the Doctor in a series of adventures and a few more stand-alone stories. McGann played an alternate version of the Eighth Doctor from a timeline in which the Seventh Doctor regenerates after being shot by Nazis in the 1950s (first mentioned in the audio drama "Colditz"), and appeared in "Klein's Story". As "Schmidt", the Doctor teaches Klein how to pilot the TARDIS and puts her in a position to inadvertently help the Seventh Doctor prevent this timeline from occurring (unable to go back himself, since the TARDIS security protocols prevent him from going somewhere he already was).


title: Eighth Doctor

text: Big Finish released a 50th-anniversary story in 2013, "The Light at the End", in which the Master (Geoffrey Beevers) implements a plot to trap and destroy all eight incarnations of the Doctor by keeping him on Gallifrey. The Eighth Doctor and Charley meet with the Fourth Doctor (Tom Baker) and Leela (Louise Jameson), the Fifth Doctor (Peter Davison) and Nyssa (Sarah Sutton), the Sixth Doctor (Colin Baker) and Peri, and the Seventh Doctor (Sylvester McCoy) and Ace (Sophie Aldred). Working with the First Doctor (William Russell), the Second Doctor (Frazer Hines), and the Third Doctor (Tim Treloar)), they escape the Master's trap and prevent time from being rewritten so the Doctor would not leave Gallifrey. Companions Susan (Carole Ann Ford), Ian Chesterton (William Russell), Vicki (Maureen O'Brien), Steven Taylor (Peter Purves), Sara Kingdom (Jean Marsh), Polly (Anneke Wills), Jamie McCrimmon (Frazer Hines), Zoe Heriot (Wendy Padbury), Jo Grant (Katy Manning), Tegan Jovanka (Janet Fielding), and Turlough (Mark Strickson) also appeared in the audio drama. In the 2013 live-action mini-episode "The Night of the Doctor", the Eighth Doctor salutes companions Charley, C'rizz, Lucie, Tamsin and Molly before he regenerates into the War Doctor.


title: Eighth Doctor

text: The Eighth Doctor's earliest audio-released adventures were a trilogy, released from October to December 2011, in which the Doctor travels with Mary Shelley (Julie Cox). Mary first appears as a friend of the Eighth Doctor in "The Company of Friends", when she meets a future version of the Doctor and helps his past self save his life. The first story, "The Silver Turk", features the early Cybermen (as seen in "The Tenth Planet"). "The Witch from the Well" tests the pair's relationship; the trilogy concludes with "Army of Death", in which Mary decides to leave the Doctor.


title: Eighth Doctor

text: In the Eighth Doctor's first audio adventure, "Storm Warning", the Doctor lands on Earth in October 1930 aboard the doomed R101 airship. On the airship the Doctor meets Charley Pollard (India Fisher), a young adventuress. He saves Charley's life, and brings her aboard the TARDIS as his latest companion. The Doctor soon learns that Charley's rescue negatively affects the timestream.


title: Eighth Doctor

text: Charley dies on the R101 in another story, and the Doctor cannot return her to preserve the timeline. Her existence forms a rough story arc over two seasons of audio adventures as the Doctor discovers a series of minor historical anomalies caused by damage to the timeline, such as the CIA's existence in 1933 or Benjamin Franklin being president. The Doctor eventually learns that, due to Charley's survival, the universe has become infected with "anti-time" leading to a conflict with the wraith-like Never People: individuals who were erased from history for crimes against reality, and now seek to unleash anti-time on the universe. The Doctor sacrifices himself and the TARDIS by absorbing anti-time energy, which seals the rift and preserves Charley's life but transforms him into the bogeyman Zagreus. Later restored to sanity but still infected with anti-time due to the actions of Charley, his old companions Leela and Romana, and recreations of his three previous incarnations, the Doctor again sacrifices himself for Charley and the universe by removing himself from space and time. He enters a divergent universe of which he has no prior knowledge (or frame of reference), and in which there is no linear time ("Zagreus"). Charley stows away on the TARDIS, negating the Doctor's sacrifice by again placing herself in danger.


title: Eighth Doctor

text: For two more seasons, the Doctor, Charley, and a new companion named C'rizz explore the divergent universe. They gradually unravel a plot designed around the Doctor by Rassilon, founder of the Time Lord society, who had been trapped in anti-time by the Doctor for his role in the creation of Zagreus. With the aid of his companions the Doctor escapes the trap, overcomes his emotional burden, learns that he has been purged of Zagreus, and returns to his normal universe with Charley and C'rizz ("The Next Life").


title: Eighth Doctor

text: After this, which coincided with the end of the Big Finish "seasons" in light of the 2005 return of "Doctor Who" to television, the trio wanders freely. The only continuing plot element involves C'rizz, who exhibits unusual (and potentially destructive) psychological development. This culminates in C'rizz's death ("Absolution"), which makes Charley want to leave the Doctor's company. Due to amnesia from a healing coma induced by a cyber-planner, the Eighth Doctor believes that Charley has left him in 2008 Singapore ("The Girl Who Never Was"); she is instead displaced in the year 500,002, from which she is rescued by the Sixth Doctor ("The Condemned"). Believing that the Eighth Doctor died before they parted, Charley travels with the Sixth Doctor. Their time together ends when the Viyrans, an alien race, replace his memories of Charley with Mila (another woman) to preserve the timeline.


title: Eighth Doctor

text: "", featuring the Doctor and new companion Lucie Miller (Sheridan Smith), began on BBC7 on New Year's Eve 2006; a second series followed in 2007. A third season was streamed on the Big Finish website, and was available on CD. A fourth (and final) season began in 2009 with the Christmas episode "Death in Blackpool", which dealt with Lucie's departure. The standalone episode "An Earthly Child" was also released in December 2009, in which the Doctor visits his granddaughter Susan Foreman on Earth (following the events of the First Doctor adventure "The Dalek Invasion of Earth") and meets his great-grandson Alex Campbell. In "Situation Vacant" the Doctor gains a new companion, Tamsin Drew, chosen from four prospective companions. This season pits the Eighth Doctor against a new incarnation of the Meddling Monk (now travelling with Lucie Miller). During the season, Tamsin is seduced to the Meddling Monk's way of thinking; Lucie finds the Monk's violence distasteful, and re-joins the Doctor. When the Monk helps the Daleks invade Earth, Tamsin, Lucie, and Alex are killed in driving them off; at the last minute, however, the Monk betrays the Daleks to help save the Doctor and Susan.


title: Eighth Doctor

text: In November 2012 Big Finish released "Dark Eyes", a box set of four audio dramas whose stories take place after "To the Death". The set introduced a new companion, Molly O'Sullivan (played by Ruth Bradley), an Irish Voluntary Aid Detachment nursing assistant in World War I. The Eighth Doctor changes his appearance, cutting his hair and changing into a World War I navy peacoat after his Victorian clothing is ruined. After the success of the first "Dark Eyes" box set, Big Finish released three more sets continuing Molly's story and introducing Liv Chenka as a new companion; Liv had met the Seventh Doctor in the audio drama "Robophobia".


title: Eighth Doctor

text: The Eighth Doctor appeared in "Doom Coalition", a four-CD, four-part miniseries. He, Liv and new companion Helen Sinclair hunt for the insane Time Lord known as the Eleven, all of whose past personalities are active in his mind at once. The Doctor interacts with River Song, the Eleventh Doctor's future wife (although she avoids interacting with the Eighth Doctor unless she is disguised), and confronts an alliance between the Monk and the Weeping Angels. "Doom Coalition" concludes with the Doctor thwarting the Doom Coalition, a group of Time Lords led by the Doctor's old schoolfriend Padrac. Padrac's latest consultations with the Matrix led him to the conclusion that the only way to save Gallifrey from prophecies of its destruction is to destroy the rest of the universe. The hunt for the Eleven is intended to manipulate the Doctor into gathering the equipment needed for Padrac's plan to succeed, but the Doctor and his companions disrupt Padrac's machines at the last minute. Big Finish began releasing two Eighth Doctor box sets in 2017: "Ravenous" and "The Eighth Doctor: The Time War".


title: Eighth Doctor

text: "Ravenous" is a series of four four-part box sets which are a sequel to "Doom Coalition", initially exploring the Doctor and Liv's efforts to find Helen after she is trapped with the Eleven. It then narrates their efforts to oppose the Ravenous, an ancient race who are predators of the Time Lords; they evolved to feed on the energy released when Time Lords regenerate. The trio briefly ally with the Eleven against the Ravenous, but he betrays them when he learns that his condition makes him immune to the Ravenous' feeding habits (prompting him to assist the Ravenous in their campaign to turn the universe into their food. The series concludes with the Doctor working with three incarnations of the Master (the Master of the TV film, the War Master and Missy) to restore the Ravenous to their original state.


title: Eighth Doctor

text: Following the events of "Ravenous", the next Eighth Doctor saga, "Stranded", sees the Doctor, Liv and Helen crash land on Earth in 2020. The Ravenous' attack on the TARDIS has made it inoperable so the trio move into the Doctor's house on Baker Street, which has since been turned into flats and already occupied by several tenants, including Torchwood agent Tania Bell. The trio attempt to adapt to life in 2020 without alien invasion or other threats while the Doctor works on repairing the TARDIS. However, they soon become aware of Divine Intervention, a self-help organisation from the future whose machinations create a divergent timeline that culminates in the destruction of the human race. When the Doctor and his friends restore the timeline, they find themselves back on Earth in the midst of the COVID-19 pandemic while waiting for the TARDIS repairs to be complete. On New Year's Day 2021, the Doctor, Liv and Helen set off once again for further adventures.


title: Eighth Doctor

text: After Big Finish brought the to a close in March 2021, the decision was made to transition all Doctor Who ranges to . Though "The Eighth Doctor Adventures" was already releasing as box sets, Big Finish decided to reformat the range further following the conclusion of "Stranded", moving away from the four-part sagas and into standalone box sets, bringing it in line with the relaunched ranges of other Doctors. The first official releases under this relaunch were two box sets - "" and "" - released in November and December 2022 respectively, and continue on from "Stranded" with Liv and Helen as the Eighth Doctor's companions.


title: Eighth Doctor

text: "The Eighth Doctor: The Time War" is another series of four four-part box sets set later in the Eighth Doctor's life, during the early days of the Time War. It describes his efforts to protect those caught in the crossfire with the aid of Bliss, his new companion, whose planet's history was undone when a Time Lord agent turned her planet into a weapons depot; her history is chaotic due to the Time War. In the third volume, the Doctor again faces the Valeyard when his other self is recreated after he is exposed to biology-manipulating technology while using a transmat. The Valeyard erases the Daleks from history by turning their latest weapon against them, but the Dalek Time Strategist escapes into another universe. In the fourth volume, the Strategist recreates the Daleks by manipulating an alternate version of Davros to combine alternate versions of Skaro (manipulating Davros into believing that the Doctor sees all Kaleds as enemies, rather than the Daleks). The Daleks rebuild their forces, but the alternate Davros is put into stasis.


title: Eighth Doctor

text: In the "Doctor Who Magazine" comic strips, at an unstated time after his regeneration (and after an adventure in the 1930s involving Fey Truscott-Sade and psychic weasels), the Doctor revisited the town of Stockbridge. After being caught up in the games of the Celestial Toymaker, he picked up a new companion in lively sci-fi fan Izzy Sinclair. The two of them were soon caught up in the machinations of the Doctor's old enemies the Threshold, a mercenary organisation. The Threshold attempted to manipulate the Doctor into stopping the Daleks gaining access to the multiverse (which would kill an artificial solar system as a side effect) and dying in the attempt, but were outmaneuvered. Unknown to him, they implanted a device in Fey Truscott-Sade so that they could use her as an unwilling spy when she next encountered the Doctor. She did so in 1939, assisting him and Izzy against the vampiric Varney; the Doctor was left infected with a deadly bacillus, and he had to be taken to Gallifrey to be cured, luring him into a battle against a Time Lord cult called the Final Chapter.


title: Eighth Doctor

text: Working out that the Threshold were using Fey as a spy, the Doctor and his old comrade Shayde faked a regeneration. The Threshold was conned into believing they were facing a vulnerable new Doctor (Shayde in disguise), allowing the real Doctor to infiltrate their base. While he and his friends were too late to stop the Threshold from destroying every single spacecraft in the universe, they were able to bring about the organisation's destruction before it could profit. Fey returned to her time, having also bonded with Shayde to save his life.


title: Eighth Doctor

text: Unknowingly, the TARDIS had been taken over by the Master, who was manipulating events to gain the power of the omniversal Glory. The Doctor was specifically sent to times and places that would undermine him – discovering he had upset the course of Grace Holloway's life in 2001, encountering an alien race with motivations uncomfortably similar to his which caused death and horror in 17th Century Japan, and almost killing the benevolent Kroton by mistake. (A slight diversion between events saw the Doctor and Izzy team up with the actor Tom Baker and other 1970s television actors against Beep the Meep in 1979.)


title: Eighth Doctor

text: In 17th Century Japan, the Doctor's attempt to save the life of samurai Katsura Sato, a friend of Izzy, left the man inadvertently immortal and thus robbed of both an honourable death and any sense of empathy. The Master later came for Sato, when he was mentally vulnerable, and gave him a fake religion to focus his mind on; Earth's history was altered as Sato, renamed Lord Morningstar, and his Church of the Glorious Dead conquered the planet, creating a technological advanced, highly brutal planet of jihadists. The Doctor, Izzy and Kroton wandered into their invasion of the museum planet Paradost; while the Doctor faced the Master over the Glory, Izzy and Kroton spent weeks on the occupied world. The Doctor was defeated, only for the Master discover he was not able to access the Glory, as instead Kroton and Sato had been the ones prophesied to battle for it. Kroton won the Glory, the Master was purged from the TARDIS and history was reverted, and the Doctor and Izzy took a well-earned break.


title: Eighth Doctor

text: During a battle against the body-stealing Ophidians and their gigantic snake-shaped techno-organic warship, the Doctor and Izzy encountered a brash and adventurous fish-woman called Destrii. While seemingly friendly and bonding with Izzy, Destrii was secretly on the run and she swapped bodies with Izzy to cover her escape; when Destrii was seemingly killed, Izzy seemed trapped within an alien body. The Doctor's next few journeys were spent trying to help her in this situation, both in coming to terms with the change and finding out what her new body's abilities were. Frida Kahlo helped Izzy mentally deal with the change, but the attempt at testing Izzy's abilities led her and the Doctor into a turbulent encounter with the humanised Daleks he had created in his second incarnation. Unable to prevent their tragic end – self-destructing to escape the machinations of the malevolent psychic Kata-Phobus -, the two of them were distracted and caught off-guard when Helioth and Hassana, two of the energy-beings called the Horde, abducted Izzy thinking she was Destrii.


title: Eighth Doctor

text: The Doctor went on a relentless search to rescue his friend, with the help of Fey/Shayde and by forcing co-operation from Destrii, still alive in her stolen body. His quest led him to the planet Oblivion, a surreal and brutal world ruled by Destrii's mother, the Matriax. Izzy was rescued and returned to her original form, while Oblivion's court system and the menace of the Horde were both destroyed, leaving Destrii free to leave her world and explore the universe with her roguish uncle Jodafra. However, Izzy had decided she wanted to return home to her family, and the Doctor was left alone.


title: Eighth Doctor

text: Feeling slightly morose, the Doctor was cheered up by an unknowing encounter with his old companion Frobisher and went on several 'holiday' adventures on his own. He eventually re-encountered Destrii and Jodafra in America during the 19th Century, where the upcoming clash between General Custer and Chief Sitting Bull was interrupted by Jodafra's machinations involving the monstrous wendigo. Jodafra had made a deal with the creature: power in exchange for being fed children. Unable to stomach this and with the Doctor urging her to listen to the spark of decency in her, Destrii helped the Doctor stop her uncle. In revenge, she was left beaten and abandoned, and the Doctor took her in as a probationary companion. Together, the two of them teamed up with MI6 and faced an invasion of early 21st Century Earth by time-travelling Cybermen; they were preparing to chemically overload the emotions of humans and thus make them willingly surrender to have their emotions removed by conversion. The Doctor destroyed them through use of the Time Vortex (similar to the later "The Parting of the Ways"), almost surrendering to it but giving up its power to save Destrii.


title: Eighth Doctor

text: In 2007, Panini Books published "Doctor Who: The Flood", the final collection of comic strips featuring the Eighth Doctor in "Doctor Who Magazine". The book includes the essay "Flood Barriers" by strip editor Clayton Hickman in which he reveals that Russell T Davies had authorised the comic strip to depict the regeneration of the Eighth Doctor into the Ninth Doctor at the end of the 2004–2005 arc, "The Flood". The cause of the regeneration would have been the Doctor's exposure to the Time Vortex in his efforts to destroy the Cybermen (the same cause that triggered the later Ninth to Tenth Doctor regeneration in "Parting of the Ways"). Destrii would have witnessed the regeneration and would have continued to travel with the Ninth Doctor in a proposed "Year One" arc. However BBC Wales vetoed the "Year One" arc and indicated the Ninth Doctor could only be shown travelling with Rose Tyler, Hickman and writer Scott Gray eventually decided not to depict the regeneration as they would have been unable to give Destrii a proper departure. The Panini collection includes the original script for the regeneration sequence, as well as never before published art showing the regeneration itself.


title: Eighth Doctor

text: In the "Doctor Who: The Eighth Doctor" comic series, the Eighth Doctor – from a point shortly before the Time War, as he is shown in the clothing and hairstyle he wore in "The Night of the Doctor" – is visiting an old holiday house of his where he encounters a young painter named Josie Day. Intrigued at her presence, he invites her to accompany him while he checks off items on a 'To Do' list left behind in his last visit to the house. In the course of the storyline, it is revealed that Josie is actually a sentient painting given life by a wealthy woman in the future as part of the woman's plan to grant herself immortality by transferring her mind into Josie, but Josie was rescued by the Twelfth Doctor and taken into the past so that she could travel with the Eighth, the Twelfth Doctor feeling that his past self was better suited to help Josie learn how to be human.


title: Basis (linear algebra)

text: In mathematics, a set of vectors in a vector space is called a basis if every element of may be written in a unique way as a finite linear combination of elements of . The coefficients of this linear combination are referred to as components or coordinates of the vector with respect to . The elements of a basis are called .


title: Basis (linear algebra)

text: Equivalently, a set is a basis if its elements are linearly independent and every element of is a linear combination of elements of . In other words, a basis is a linearly independent spanning set.


title: Basis (linear algebra)

text: A vector space can have several bases; however all the bases have the same number of elements, called the "dimension" of the vector space.


title: Basis (linear algebra)

text: This article deals mainly with finite-dimensional vector spaces. However, many of the principles are also valid for infinite-dimensional vector spaces.


title: Basis (linear algebra)

text: A basis of a vector space over a field (such as the real numbers or the complex numbers ) is a linearly independent subset of that spans . This means that a subset of is a basis if it satisfies the two following conditions:


title: Basis (linear algebra)

text: The scalars formula_6 are called the coordinates of the vector with respect to the basis , and by the first property they are uniquely determined.


title: Basis (linear algebra)

text: A vector space that has a finite basis is called finite-dimensional. In this case, the finite subset can be taken as itself to check for linear independence in the above definition.


title: Basis (linear algebra)

text: It is often convenient or even necessary to have an ordering on the basis vectors, for example, when discussing orientation, or when one considers the scalar coefficients of a vector with respect to a basis without referring explicitly to the basis elements. In this case, the ordering is necessary for associating each coefficient to the corresponding basis element. This ordering can be done by numbering the basis elements. In order to emphasize that an order has been chosen, one speaks of an ordered basis, which is therefore not simply an unstructured set, but a sequence, an indexed family, or similar; see below.


title: Basis (linear algebra)

text: The set of the ordered pairs of real numbers is a vector space under the operations of component-wise addition


title: Basis (linear algebra)

text: where formula_9 is any real number. A simple basis of this vector space consists of the two vectors and . These vectors form a basis (called the standard basis) because any vector of may be uniquely written as formula_10 Any other pair of linearly independent vectors of , such as and , forms also a basis of .


title: Basis (linear algebra)

text: More generally, if is a field, the set formula_11 of -tuples of elements of is a vector space for similarly defined addition and scalar multiplication. Let formula_12 be the -tuple with all components equal to 0, except the th, which is 1. Then formula_13 is a basis of formula_14 which is called the "standard basis" of formula_15


title: Basis (linear algebra)

text: A different flavor of example is given by polynomial rings. If is a field, the collection of all polynomials in one indeterminate with coefficients in is an -vector space. One basis for this space is the monomial basis , consisting of all monomials: formula_16 Any set of polynomials such that there is exactly one polynomial of each degree (such as the Bernstein basis polynomials or Chebyshev polynomials) is also a basis. (Such a set of polynomials is called a polynomial sequence.) But there are also many bases for that are not of this form.


title: Basis (linear algebra)

text: Many properties of finite bases result from the Steinitz exchange lemma, which states that, for any vector space , given a finite spanning set and a linearly independent set of elements of , one may replace well-chosen elements of by the elements of to get a spanning set containing , having its other elements in , and having the same number of elements as .


title: Basis (linear algebra)

text: Most properties resulting from the Steinitz exchange lemma remain true when there is no finite spanning set, but their proofs in the infinite case generally require the axiom of choice or a weaker form of it, such as the ultrafilter lemma.


title: Basis (linear algebra)

text: where the coefficients formula_20 are scalars (that is, elements of ), which are called the "coordinates" of over . However, if one talks of the "set" of the coefficients, one loses the correspondence between coefficients and basis elements, and several vectors may have the same "set" of coefficients. For example, formula_21 and formula_22 have the same set of coefficients , and are different. It is therefore often convenient to work with an ordered basis; this is typically done by indexing the basis elements by the first natural numbers. Then, the coordinates of a vector form a sequence similarly indexed, and a vector is completely characterized by the sequence of coordinates. An ordered basis is also called a frame, a word commonly used, in various contexts, for referring to a sequence of data allowing defining coordinates.


title: Basis (linear algebra)

text: Let, as usual, formula_11 be the set of the -tuples of elements of . This set is an -vector space, with addition and scalar multiplication defined component-wise. The map


title: Basis (linear algebra)

text: is a linear isomorphism from the vector space formula_11 onto . In other words, formula_11 is the coordinate space of , and the -tuple formula_27 is the coordinate vector of .


title: Basis (linear algebra)

text: The inverse image by formula_28 of formula_29 is the -tuple formula_30 all of whose components are 0, except the th that is 1. The formula_30 form an ordered basis of formula_11, which is called its standard basis or canonical basis. The ordered basis is the image by formula_28 of the canonical basis of


title: Basis (linear algebra)

text: It follows from what precedes that every ordered basis is the image by a linear isomorphism of the canonical basis of and that every linear isomorphism from formula_11 onto may be defined as the isomorphism that maps the canonical basis of formula_11 onto a given ordered basis of . In other words it is equivalent to define an ordered basis of , or a linear isomorphism from formula_11 onto .


title: Basis (linear algebra)

text: Let be a vector space of dimension over a field . Given two (ordered) bases formula_37 and formula_38 of , it is often useful to express the coordinates of a vector with respect to formula_39 in terms of the coordinates with respect to formula_40 This can be done by the "change-of-basis formula", that is described below. The subscripts "old" and "new" have been chosen because it is customary to refer to formula_39 and formula_42 as the "old basis" and the "new basis", respectively. It is useful to describe the old coordinates in terms of the new ones, because, in general, one has expressions involving the old coordinates, and if one wants to obtain equivalent expressions in terms of the new coordinates; this is obtained by replacing the old coordinates by their expressions in terms of the new coordinates.


title: Basis (linear algebra)

text: If formula_44 and formula_45 are the coordinates of a vector over the old and the new basis respectively, the change-of-basis formula is


title: Basis (linear algebra)

text: be the column vectors of the coordinates of in the old and the new basis respectively, then the formula for changing coordinates is


title: Basis (linear algebra)

text: The change-of-basis formula results then from the uniqueness of the decomposition of a vector over a basis, here that is


title: Basis (linear algebra)

text: If one replaces the field occurring in the definition of a vector space by a ring, one gets the definition of a module. For modules, linear independence and spanning sets are defined exactly as for vector spaces, although "generating set" is more commonly used than that of "spanning set".


title: Basis (linear algebra)

text: Like for vector spaces, a "basis" of a module is a linearly independent subset that is also a generating set. A major difference with the theory of vector spaces is that not every module has a basis. A module that has a basis is called a "free module". Free modules play a fundamental role in module theory, as they may be used for describing the structure of non-free modules through free resolutions.


title: Basis (linear algebra)

text: A module over the integers is exactly the same thing as an abelian group. Thus a free module over the integers is also a free abelian group. Free abelian groups have specific properties that are not shared by modules over other rings. Specifically, every subgroup of a free abelian group is a free abelian group, and, if is a subgroup of a finitely generated free abelian group (that is an abelian group that has a finite basis), then there is a basis formula_13 of and an integer such that formula_53 is a basis of , for some nonzero integers For details, see .


title: Basis (linear algebra)

text: In the context of infinite-dimensional vector spaces over the real or complex numbers, the term (named after Georg Hamel) or algebraic basis can be used to refer to a basis as defined in this article. This is to make a distinction with other notions of "basis" that exist when infinite-dimensional vector spaces are endowed with extra structure. The most important alternatives are orthogonal bases on Hilbert spaces, Schauder bases, and Markushevich bases on normed linear spaces. In the case of the real numbers R viewed as a vector space over the field Q of rational numbers, Hamel bases are uncountable, and have specifically the cardinality of the continuum, which is the cardinal number where formula_54 is the smallest infinite cardinal, the cardinal of the integers.


title: Basis (linear algebra)

text: The common feature of the other notions is that they permit the taking of infinite linear combinations of the basis vectors in order to generate the space. This, of course, requires that infinite sums are meaningfully defined on these spaces, as is the case for topological vector spaces – a large class of vector spaces including e.g. Hilbert spaces, Banach spaces, or Fréchet spaces.


title: Basis (linear algebra)

text: The preference of other types of bases for infinite-dimensional spaces is justified by the fact that the Hamel basis becomes "too big" in Banach spaces: If "X" is an infinite-dimensional normed vector space which is complete (i.e. "X" is a Banach space), then any Hamel basis of "X" is necessarily uncountable. This is a consequence of the Baire category theorem. The completeness as well as infinite dimension are crucial assumptions in the previous claim. Indeed, finite-dimensional spaces have by definition finite bases and there are infinite-dimensional ("non-complete") normed spaces which have countable Hamel bases. Consider the space of the sequences formula_55 of real numbers which have only finitely many non-zero elements, with the norm Its standard basis, consisting of the sequences having only one non-zero element, which is equal to 1, is a countable Hamel basis.


title: Basis (linear algebra)

text: In the study of Fourier series, one learns that the functions are an "orthogonal basis" of the (real or complex) vector space of all (real or complex valued) functions on the interval [0, 2π] that are square-integrable on this interval, i.e., functions "f" satisfying


title: Basis (linear algebra)

text: The functions are linearly independent, and every function "f" that is square-integrable on [0, 2π] is an "infinite linear combination" of them, in the sense that


title: Basis (linear algebra)

text: for suitable (real or complex) coefficients "a", "b". But many square-integrable functions cannot be represented as "finite" linear combinations of these basis functions, which therefore "do not" comprise a Hamel basis. Every Hamel basis of this space is much bigger than this merely countably infinite set of functions. Hamel bases of spaces of this kind are typically not useful, whereas orthonormal bases of these spaces are essential in Fourier analysis.


title: Basis (linear algebra)

text: The geometric notions of an affine space, projective space, convex set, and cone have related notions of "basis". An affine basis for an "n"-dimensional affine space is formula_58 points in general linear position. A ' is formula_59 points in general position, in a projective space of dimension "n". A ' of a polytope is the set of the vertices of its convex hull. A consists of one point by edge of a polygonal cone. See also a Hilbert basis (linear programming).


title: Basis (linear algebra)

text: For a probability distribution in with a probability density function, such as the equidistribution in an "n"-dimensional ball with respect to Lebesgue measure, it can be shown that randomly and independently chosen vectors will form a basis with probability one, which is due to the fact that linearly dependent vectors , ..., in should satisfy the equation (zero determinant of the matrix with columns ), and the set of zeros of a non-trivial polynomial has zero measure. This observation has led to techniques for approximating random bases.


title: Basis (linear algebra)

text: It is difficult to check numerically the linear dependence or exact orthogonality. Therefore, the notion of ε-orthogonality is used. For spaces with inner product, "x" is ε-orthogonal to "y" if formula_60 (that is, cosine of the angle between and is less than ).


title: Basis (linear algebra)

text: In high dimensions, two independent random vectors are with high probability almost orthogonal, and the number of independent random vectors, which all are with given high probability pairwise almost orthogonal, grows exponentially with dimension. More precisely, consider equidistribution in "n"-dimensional ball. Choose "N" independent random vectors from a ball (they are independent and identically distributed). Let "θ" be a small positive number. Then for


title: Basis (linear algebra)

text: The figure (right) illustrates distribution of lengths N of pairwise almost orthogonal chains of vectors that are independently randomly sampled from the "n"-dimensional cube as a function of dimension, "n". A point is first randomly selected in the cube. The second point is randomly chosen in the same cube. If the angle between the vectors was within then the vector was retained. At the next step a new vector is generated in the same hypercube, and its angles with the previously generated vectors are evaluated. If these angles are within then the vector is retained. The process is repeated until the chain of almost orthogonality breaks, and the number of such pairwise almost orthogonal vectors (length of the chain) is recorded. For each "n", 20 pairwise almost orthogonal chains were constructed numerically for each dimension. Distribution of the length of these chains is presented.


title: Basis (linear algebra)

text: The set is nonempty since the empty set is an independent subset of , and it is partially ordered by inclusion, which is denoted, as usual, by .


title: Basis (linear algebra)

text: Let be a subset of that is totally ordered by , and let be the union of all the elements of (which are themselves certain subsets of ).


title: Basis (linear algebra)

text: Since is totally ordered, every finite subset of is a subset of an element of , which is a linearly independent subset of , and hence is linearly independent. Thus is an element of . Therefore, is an upper bound for in : it is an element of , that contains every element of .


title: Basis (linear algebra)

text: As is nonempty, and every totally ordered subset of has an upper bound in , Zorn's lemma asserts that has a maximal element. In other words, there exists some element of satisfying the condition that whenever for some element of , then .


title: Basis (linear algebra)

text: It remains to prove that is a basis of . Since belongs to , we already know that is a linearly independent subset of .


title: Basis (linear algebra)

text: If there were some vector of that is not in the span of , then would not be an element of either. Let . This set is an element of , that is, it is a linearly independent subset of (because w is not in the span of L, and is independent). As , and (because contains the vector that is not contained in ), this contradicts the maximality of . Thus this shows that spans .


title: Basis (linear algebra)

text: Hence is linearly independent and spans . It is thus a basis of , and this proves that every vector space has a basis.


title: Basis (linear algebra)

text: This proof relies on Zorn's lemma, which is equivalent to the axiom of choice. Conversely, it has been proved that if every vector space has a basis, then the axiom of choice is true. Thus the two assertions are equivalent.


title: Ames, Iowa

text: Ames () is a city in Story County, Iowa, United States, located approximately north of Des Moines in central Iowa. It is best known as the home of Iowa State University (ISU), with leading agriculture, design, engineering, and veterinary medicine colleges. A United States Department of Energy national laboratory, Ames Laboratory, is located on the ISU campus.


title: Ames, Iowa

text: According to the 2020 census, Ames had a population of 66,427, making it the state's ninth largest city. Iowa State University was home to 33,391 students as of fall 2019, which make up approximately one half of the city's population.


title: Ames, Iowa

text: Ames also hosts United States Department of Agriculture (USDA) sites: the largest federal animal disease center in the United States, the USDA Agricultural Research Service's National Animal Disease Center (NADC), as well as one of two national USDA sites for the Animal and Plant Health Inspection Service (APHIS), which comprises the National Veterinary Services Laboratory and the Center for Veterinary Biologics. Ames also hosts the headquarters for the Iowa Department of Transportation.


title: Ames, Iowa

text: The city was formed in 1864 as a station stop on the Cedar Rapids and Missouri Railroad and was named after 19th century U.S. Congressman Oakes Ames of Massachusetts, who was influential in the building of the transcontinental railroad. Ames was founded by local resident Cynthia Olive Duff (née Kellogg) and railroad magnate John Insley Blair, near a location that was deemed favorable for a railroad crossing of the Skunk River.


title: Ames, Iowa

text: Ames is located along the western edge of Story County, roughly north of the state capital, Des Moines, near the intersection of Interstate 35 and U.S. Route 30. A smaller highway, U.S. Route 69, passes through the town. Also passing through Ames is the cross country line of the Union Pacific Railroad and two small streams (the South Skunk River and Ioway Creek).


title: Ames, Iowa

text: According to the United States Census Bureau, the city has a total area of , of which is land and is water.


title: Ames, Iowa

text: Campustown is the neighborhood directly south of Iowa State University Central Campus bordered by Lincoln Way on the north. Campustown is a high-density mixed-use neighborhood that is home to many student apartments, nightlife venues, restaurants, and numerous other establishments, most of which are unique to Ames.


title: Ames, Iowa

text: Ames has a humid continental climate (Köppen climate classification "Dfa"). On average, the warmest month is July and the coldest is January. The highest recorded temperature was in 1988 and the lowest was in 1996.


title: Ames, Iowa

text: As of the census of 2010, there were 58,965 people, 22,759 households, and 9,959 families residing in the city. The population density was . There were 23,876 housing units at an average density of . The racial makeup of the city was 84.5% White, 3.4% African American, 0.2% Native American, 8.8% Asian, 1.1% from other races, and 2.0% from two or more races. Hispanic or Latino of any race were 3.4% of the population.


title: Ames, Iowa

text: There were 22,759 households, of which 19.1% had children under the age of 18 living with them, 35.6% were married couples living together, 5.4% had a female householder with no husband present, 2.7% had a male householder with no wife present, and 56.2% were non-families. 30.5% of all households were made up of individuals, and 6.2% had someone living alone who was 65 years of age or older. The average household size was 2.25 and the average family size was 2.82.


title: Ames, Iowa

text: The median age in the city was 23.8 years. 13.4% of residents were under the age of 18; 40.5% were between the ages of 18 and 24; 22.9% were from 25 to 44; 15% were from 45 to 64; and 8.1% were 65 years of age or older. The gender makeup of the city was 53.0% male and 47.0% female.


title: Ames, Iowa

text: As of the census of 2000, there were 50,731 people, 18,085 households, and 8,970 families residing in the city. The population density was . There were 18,757 housing units at an average density of . The racial makeup of the city was 87.34% White, 7.70% Asian, 2.65% African American, 0.04% Native American, 0.76% Pacific Islander and other races, and 1.36% from two or more races. Hispanic or Latino of any race were 1.98% of the population.


title: Ames, Iowa

text: There were 18,085 households, out of which 22.3% had children under the age of 18 living with them, 42.0% were married couples living together, 5.3% had a female householder with no husband present, and 50.4% were non-families. 28.5% of all households were made up of individuals, and 5.9% had someone living alone who was 65 years of age or older. The average household size was 2.30 and the average family size was 2.85.


title: Ames, Iowa

text: Age spread: 14.6% under the age of 18, 40.0% from 18 to 24, 23.7% from 25 to 44, 13.9% from 45 to 64, and 7.7% who were 65 years of age or older. The median age was 24 years. For every 100 females, there were 109.3 males. For every 100 females age 18 and over, there were 109.9 males.


title: Ames, Iowa

text: The median income for a household in the city was $36,042, and the median income for a family was $56,439. Males had a median income of $37,877 versus $28,198 for females. The per capita income for the city was $18,881. About 7.6% of families and 20.4% of the population were below the poverty line, including 9.2% of those under age 18 and 4.1% of those age 65 or over.


title: Ames, Iowa

text: The U.S. Census Bureau designates the Ames MSA as encompassing all of Story County. While Ames is the largest city in Story County, the county seat is in the nearby city of Nevada, east of Ames.


title: Ames, Iowa

text: Ames metropolitan statistical area combined with the Boone, Iowa micropolitan statistical area (Boone County, Iowa) make up the larger Ames-Boone combined statistical area. Ames is the larger principal city of the Combined Statistical Area that includes all of Story County, Iowa and Boone County, Iowa. which had a combined population of 106,205 at the 2000 census.


title: Ames, Iowa

text: Ames is home of Iowa State University of Science and Technology, a public land-grant and space-grant research university. At its founding in 1858, Iowa State was formerly known as the Iowa State College of Agriculture and Mechanic Arts. Ames is the home of the closely allied U.S. Department of Agriculture's National Animal Disease Center (See Ames strain), the U.S. Department of Energy's Ames Laboratory (a major materials research and development facility), and the main offices of the Iowa Department of Transportation. State and Federal institutions are the largest employers in Ames.


title: Ames, Iowa

text: Other area employers include a 3M manufacturing plant; Danfoss Power Solutions, a hydraulics manufacturer; Barilla, a pasta manufacturer; Ball, a manufacturer of canning jars and plastic bottles; Workiva, a global cloud computing company; Renewable Energy Group, America's largest producer of biomass-based diesel; and the National Farmers Organization.


title: Ames, Iowa

text: The Iowa State University Research Park is a not-for-profit, business development incubator located in Ames, and affiliated with Iowa State University.


title: Ames, Iowa

text: In 2015, Ames was ranked in the top 15 "Cities That Have Done the Best Since the Recession" by Bloomberg Businessweek.


title: Ames, Iowa

text: The Bureau of Labor Statistics ranked Ames and Boulder, CO as having the lowest unemployment rate (2.5%) of any metropolitan area in the US in 2016. By June 2018, unemployment in Ames had fallen even further, to 1.5%, and wage increases for workers were not keeping pace with rising rents.


title: Ames, Iowa

text: The Iowa State Cyclones play a variety of sports in the Ames area. The Iowa State Cyclones football team plays at Jack Trice Stadium in Ames. Also, the Cyclones' Men's and Women's Basketball teams and Volleyball teams play at Hilton Coliseum just across the street from Jack Trice Stadium. The Iowa State Cyclones are a charter member of the Big 12 Conference in all sports and compete in NCAA Division I-A. The Iowa State Women's Tennis is also well known and very successful.


title: Ames, Iowa

text: The Ames Figure Skating Club provides recreational to professional level skating opportunities. The club sponsors the Learn to Skate Program. Coaches provide on and off ice lessons or workshops. The club hosts the figure skating portion of the Iowa Games competition every summer. In the fall the club hosts Cyclone Country Championships.


title: Ames, Iowa

text: The Ames ISU ice arena also hosts the Iowa State Cyclones hockey team. The arena also hosts the Ames Little Cyclones hockey program for high school students and children in elementary or middle school.


title: Ames, Iowa

text: Iowa State University of Science and Technology, more commonly known as Iowa State University (ISU), is a public land-grant and space-grant research university located in Ames. Iowa State University is the birthplace of the Atanasoff–Berry Computer, the world's first electronic digital computer. Iowa State has produced a number of astronauts, scientists, Nobel laureates, and Pulitzer Prize winners. Until 1945 it was known as the Iowa State College of Agriculture and Mechanic Arts. The university is a member of the American Association of Universities and the Big 12 Conference.


title: Ames, Iowa

text: ISU is the nation's first designated land-grant university In 1856, the Iowa General Assembly enacted legislation to establish the State Agricultural College and Model Farm. Story County was chosen as the location on June 21, 1859, from proposals by Johnson, Kossuth, Marshall, Polk, and Story counties. When Iowa accepted the provisions of the Morrill Act of 1862, Iowa State became the first institution in nation designated as a land-grant college. The institution was coeducational from the first preparatory class admitted in 1868. The formal admitting of students began the following year, and the first graduating class of 1872 consisted of 24 men and 2 women.


title: Ames, Iowa

text: The first building on the Iowa State campus was Farm House. Built in the 1860s, it currently serves as a museum and National Historic Landmark. Today, Iowa State has over 60 notable buildings, including Beardshear Hall, Morrill Hall, Memorial Union, Catt Hall, Curtiss Hall, Carver Hall, Parks Library, the Campanile, Hilton Coliseum, C.Y. Stephens Auditorium, Fisher Theater, Jack Trice Stadium, Lied Recreation Center, numerous residence halls, and many buildings specific to ISU's many different majors and colleges.


title: Ames, Iowa

text: The official mascot for ISU is Cy the Cardinal. The official school colors are cardinal and gold. The Iowa State Cyclones play in the NCAA's Division I-A as a member of the Big 12 Conference.


title: Ames, Iowa

text: Ames is also served by stations in the Des Moines media market, which includes Clear Channel's 50,000-watt talk station WHO, music stations KAZR, KDRB, KGGO, KKDM, KHKI, KIOA, KJJY, KRNT, KSPZ and KSTZ, talk station KWQW, and sports stations KXNO and KXNO-FM.


title: Ames, Iowa

text: Like radio, Ames is served by the Des Moines media market. WOI-DT, the ABC affiliate in central Iowa, was originally owned and operated by Iowa State University until the 1990s. The station is still licensed to Ames, but studio's are located in West Des Moines. Other stations serving Ames include KCCI, KDIN-TV, WHO-DT, KCWI-TV, KDMI, KDSM-TV and KFPX-TV.


title: Ames, Iowa

text: The town is served by U.S. Highways 30 and 69 and Interstate 35. Ames is the only town in Iowa with a population of greater than 50,000 that does not have a state highway serving it. , Ames currently has three roundabouts constructed on University Avenue/530th Avenue. The first is at the intersection of Airport Road (Oakwood Rd.) and University Avenue, the second at the intersection of Cottonwood Road and 530th Avenue and the third at Collaboration Place and 530th Avenue.


title: Ames, Iowa

text: Ames was serviced by the Fort Dodge, Des Moines and Southern Railroad via a branch from Kelley to Iowa State and to downtown Ames. The tracks were removed in the 1960s. The Chicago and North Western Transportation Company twin mainline runs east and west bisecting the town and running just south of the downtown business district. The C&NW used to operate a branch to Des Moines. This line was removed in the 1980s when the Spine Line through the nearby city of Nevada was purchased from the Rock Island Railroad after its bankruptcy. The Union Pacific, successor to the C&NW, still runs 60–70 trains a day through Ames on twin mainlines, which leads to some traffic delays. There is also a branch to Eagle Grove that leaves Ames to the north. The Union Pacific maintains a small yard called Ames Yard east of Ames between Ames and Nevada. Ames has been testing automatic train horns at several of its crossings. These directional horns which are focused down the streets are activated when the crossing signals turn on and are shut off after the train crosses the crossing. This system cancels out the need for the trains to blow their horns. Train noise had been a problem in the residential areas to the west and northwest of downtown.


title: Ames, Iowa

text: Ames Municipal Airport is located southeast of the city. The current (and only) fixed-base operator is Central Iowa Air Service. The airport has two runways – 01/19, which is , and 13/31, which is .


title: Ames, Iowa

text: The City of Ames offers a transit system throughout town, called CyRide, that is funded jointly by Iowa State University, the ISU Government of the Student Body, and the City of Ames. Rider fares are free for children under five, while students pay a set cost as part of their tuition. In addition to local transit, Ames is served by intercity buses from Jefferson Lines, which stop at the Ames Intermodal Facility.


title: Ames, Iowa

text: In 2009, the Ames metropolitan statistical area (MSA) ranked as the third highest in the United States for percentage of commuters who walked to work (10.4 percent).


title: Ames, Iowa

text: Ames is served by Mary Greeley Medical Center, a 220-bed regional referral hospital which is adjacent to McFarland Clinic PC, central Iowa's largest physician-owned multi-specialty clinic, and also Iowa Heart Center.


title: Ames, Iowa

text: On September 10, 2019 the City of Ames proposed a $29,000,000 bond for building a fitness center called the Healthy Life Center. It failed to pass. Iowa State University owns the land it was to be built on.


title: Ames, Iowa

text: This is a list of notable people associated with Ames, Iowa arranged by career and in alphabetical order.


title: Ames, Iowa

text: Iowa is a political "battleground state" that has trended slightly Republican in recent years, and Ames, like Iowa City, trends Democratic. Iowa is the first caucus state and Ames is a college town. It is the site of many political appearances, debates and events, especially during election years.


title: Ames, Iowa

text: From 1979 through 2011, Ames was the location of the Ames Straw Poll, which was held every August prior to a presidential election year in which the Republican presidential nomination was undecided (meaning there was no Republican president running for re-election—as in 2011, 2007, 1999, 1995, 1987, and 1979). The poll would gauge support for the various Republican candidates amongst attendees of a fundraising dinner benefiting the Iowa Republican Party. The straw poll was frequently seen by national media and party insiders as a first test of organizational strength in Iowa. In 2015, the straw poll was to be moved to nearby Boone before the Iowa Republican Party eventually decided to cancel it altogether.


title: Solid angle

text: In geometry, a solid angle (symbol: ) is a measure of the amount of the field of view from some particular point that a given object covers. That is, it is a measure of how large the object appears to an observer looking from that point.


title: Solid angle

text: The point from which the object is viewed is called the "apex" of the solid angle, and the object is said to "subtend" its solid angle at that point.


title: Solid angle

text: In the International System of Units (SI), a solid angle is expressed in a dimensionless unit called a "steradian" (symbol: sr). One steradian corresponds to one unit of area on the unit sphere surrounding the apex, so an object that blocks all rays from the apex would cover a number of steradians equal to the total surface area of the unit sphere, formula_1. Solid angles can also be measured in squares of angular measures such as degrees, minutes, and seconds.


title: Solid angle

text: A small object nearby may subtend the same solid angle as a larger object farther away. For example, although the Moon is much smaller than the Sun, it is also much closer to Earth. Indeed, as viewed from any point on Earth, both objects have approximately the same solid angle as well as apparent size. This is evident during a solar eclipse.


title: Solid angle

text: An object's solid angle in steradians is equal to the area of the segment of a unit sphere, centered at the apex, that the object covers. Giving the area of a segment of a unit sphere in steradians is analogous to giving the length of an arc of a unit circle in radians. Just like a planar angle in radians is the ratio of the length of an arc to its radius, a solid angle in steradians is the ratio of the area covered on a sphere by an object to the area given by the square of the radius of said sphere. The formula is


title: Solid angle

text: Solid angles are often used in astronomy, physics, and in particular astrophysics. The solid angle of an object that is very far away is roughly proportional to the ratio of area to squared distance. Here "area" means the area of the object when projected along the viewing direction.


title: Solid angle

text: The solid angle of a sphere measured from any point in its interior is 4 sr, and the solid angle subtended at the center of a cube by one of its faces is one-sixth of that, or  sr. Solid angles can also be measured in square degrees (1 sr = square degrees), in square minutes and square seconds, or in fractions of the sphere (1 sr = fractional area), also known as spat (1 sp = 4 sr).


title: Solid angle

text: The solid angle for an arbitrary oriented surface subtended at a point is equal to the solid angle of the projection of the surface to the unit sphere with center , which can be calculated as the surface integral:


title: Solid angle

text: where formula_5 is the unit vector corresponding to formula_6, the position vector of an infinitesimal area of surface with respect to point , and where formula_7 represents the unit normal vector to . Even if the projection on the unit sphere to the surface is not isomorphic, the multiple folds are correctly considered according to the surface orientation described by the sign of the scalar product formula_8.


title: Solid angle

text: Thus one can approximate the solid angle subtended by a small facet having flat surface area , orientation formula_9, and distance from the viewer as:


title: Solid angle

text: The solid angle of a cone with its apex at the apex of the solid angle, and with apex angle 2, is the area of a spherical cap on a unit sphere


title: Solid angle

text: The above is found by computing the following double integral using the unit surface element in spherical coordinates:


title: Solid angle

text: This formula can also be derived without the use of calculus. Over 2200 years ago Archimedes proved that the surface area of a spherical cap is always equal to the area of a circle whose radius equals the distance from the rim of the spherical cap to the point where the cap's axis of symmetry intersects the cap. In the diagram this radius is given as


title: Solid angle

text: This is also the solid angle of the part of the celestial sphere that an astronomical observer positioned at latitude can see as the Earth rotates. At the equator all of the celestial sphere is visible; at either pole, only one half.


title: Solid angle

text: The solid angle subtended by a segment of a spherical cap cut by a plane at angle from the cone's axis and passing through the cone's apex can be calculated by the formula


title: Solid angle

text: For example, if , then the formula reduces to the spherical cap formula above: the first term becomes , and the second .


title: Solid angle

text: Let OABC be the vertices of a tetrahedron with an origin at O subtended by the triangular face ABC where formula_17 are the vector positions of the vertices A, B and C. Define the vertex angle to be the angle BOC and define , correspondingly. Let formula_18 be the dihedral angle between the planes that contain the tetrahedral faces OAC and OBC and define formula_19, formula_20 correspondingly. The solid angle subtended by the triangular surface ABC is given by


title: Solid angle

text: This follows from the theory of spherical excess and it leads to the fact that there is an analogous theorem to the theorem that ""The sum of internal angles of a planar triangle is equal to "", for the sum of the four internal solid angles of a tetrahedron as follows:


title: Solid angle

text: where formula_23 ranges over all six of the dihedral angles between any two planes that contain the tetrahedral faces OAB, OAC, OBC and ABC.


title: Solid angle

text: A useful formula for calculating the solid angle of the tetrahedron at the origin O that is purely a function of the vertex angles , , is given by L'Huilier's theorem as


title: Solid angle

text: Another interesting formula involves expressing the vertices as vectors in 3 dimensional space. Let formula_17 be the vector positions of the vertices A, B and C, and let , , and be the magnitude of each vector (the origin-point distance). The solid angle subtended by the triangular surface ABC is:


title: Solid angle

text: Care must be taken here to avoid negative or incorrect solid angles. One source of potential errors is that the scalar triple product can be negative if , , have the wrong winding. Computing is a sufficient solution since no other portion of the equation depends on the winding. The other pitfall arises when the scalar triple product is positive but the divisor is negative. In this case returns a negative value that must be increased by .


title: Solid angle

text: The solid angle of a four-sided right rectangular pyramid with apex angles and (dihedral angles measured to the opposite side faces of the pyramid) is


title: Solid angle

text: If both the side lengths ( and ) of the base of the pyramid and the distance () from the center of the base rectangle to the apex of the pyramid (the center of the sphere) are known, then the above equation can be manipulated to give


title: Solid angle

text: The solid angle of a right -gonal pyramid, where the pyramid base is a regular -sided polygon of circumradius , with a


title: Solid angle

text: The solid angle of an arbitrary pyramid with an -sided base defined by the sequence of unit vectors representing edges can be efficiently computed by:


title: Solid angle

text: where parentheses (* *) is a scalar product and square brackets [* * *] is a scalar triple product, and is an imaginary unit. Indices are cycled: and . The complex products add the phase associated with each vertex angle of the polygon. However, a multiple of


title: Solid angle

text: formula_34 is lost in the branch cut of formula_35 and must be kept track of separately. Also, the running product of complex phases must scaled occasionally to avoid underflow in the limit of nearly parallel segments.


title: Solid angle

text: where and are north and south lines of latitude (measured from the equator in radians with angle increasing northward), and and are east and west lines of longitude (where the angle in radians increases eastward). Mathematically, this represents an arc of angle swept around a sphere by radians. When longitude spans 2 radians and latitude spans radians, the solid angle is that of a sphere.


title: Solid angle

text: A latitude-longitude rectangle should not be confused with the solid angle of a rectangular pyramid. All four sides of a rectangular pyramid intersect the sphere's surface in great circle arcs. With a latitude-longitude rectangle, only lines of longitude are great circle arcs; lines of latitude are not.


title: Solid angle

text: By using the definition of angular diameter, the formula for the solid angle of a celestial object can be defined in terms of the radius of the object, formula_37, and the distance from the observer to the object, formula_38:


title: Solid angle

text: By inputting the appropriate average values for the Sun and the Moon (in relation to Earth), the average solid angle of the Sun is steradians and the average solid angle of the Moon is steradians. In terms of the total celestial sphere, the Sun and the Moon subtend average "fractional areas" of % () and % (), respectively. As these solid angles are about the same size, the Moon can cause both total and annular solar eclipses depending on the distance between the Earth and the Moon during the eclipse.


title: Solid angle

text: The solid angle subtended by the complete ()-dimensional spherical surface of the unit sphere in -dimensional Euclidean space can be defined in any number of dimensions . One often needs this solid angle factor in calculations with spherical symmetry. It is given by the formula


title: Solid angle

text: where is the gamma function. When is an integer, the gamma function can be computed explicitly. It follows that


title: Solid angle

text: This gives the expected results of 4 steradians for the 3D sphere bounded by a surface of area and 2 radians for the 2D circle bounded by a circumference of length . It also gives the slightly less obvious 2 for the 1D case, in which the origin-centered 1D "sphere" is the interval and this is bounded by two limiting points.


title: Solid angle

text: Given unit vectors formula_43 defining the angle, let denote the matrix formed by combining them so the th column is formula_43, and formula_45. The variables formula_46 form a multivariable formula_47. For a "congruent" integer multiexponent formula_48 define formula_49. Note that here formula_50 = non-negative integers, or natural numbers beginning with 0. The notation formula_51 for formula_52 means the variable formula_53, similarly for the exponents formula_54.


title: Solid angle

text: Hence, the term formula_55 means the sum over all terms in formula_56 in which l appears as either the first or second index.


title: ZIL-130

text: The ZIL-130 is a Soviet/Russian truck produced by ZIL in Moscow, Russia. The first prototype was built in 1956. Production began in 1962, while mass production started in 1964. It was one of the most numerous cargo trucks in the USSR and Russia, in total ZIL built 3,380,000 trucks up to 1994. In 1995, production was moved to the now-defunct Ural Motor Plant (UamZ, its trucks were known as UamZ-43140).


title: ZIL-130

text: The ZiL plant started working on a replacement for the ZIL-164, right after De-Stalinization took place. The first prototype was built some months later and had a brand new cabin, as well as a wider windshield influenced by Dodge trucks.


title: ZIL-130

text: The new model retained little from its predecessor, with a new V8 engine and a more reinforced frame, however, the newer truck was slightly shorter than the ZIL-164 truck. Mass-production started in 1964 under the "ZIL-130" name, and soon the ZIL-164 was discontinued in favor of its more modern successor.


title: ZIL-130

text: The ZIL-130 received the latest features adopted by the global car industry of the 1950s while not being based on any foreign model and having a unique chassis, cabin and other parts. In 1965 the heavier ZIL-131 truck started getting produced.


title: ZIL-130

text: In 1986 the company started producing the newer ZIL-4331 truck, but production of the ZIL-130 continued even after the dissolution of the Soviet Union, eventually ending in 1995, when the design tooling was sold to UamZ and production continued as the UamZ-43140 truck until 2012.


title: ZIL-130

text: Over the course of production, there were two modernizations of the ZIL-130 in 1966 and 1977. After the second one the radiator enclosure was changed.


title: ZIL-130

text: Since 1986, in accordance with the branch standard OST 37.001-269-83, the ZIL-130 series received new indexes: ZIL-431410 (ZIL-130), ZIL-431510 (ZIL-130Г), ZIL-441510 (ZIL-130В1), ZIL-431810 (ZIL-138), ZIL-431610 (ZIL-138А), etc.


title: Cross

text: A cross is a geometrical figure consisting of two intersecting lines or bars, usually perpendicular to each other. The lines usually run vertically and horizontally. A cross of oblique lines, in the shape of the Latin letter X, is termed a saltire in heraldic terminology.


title: Cross

text: The cross in its various shapes and forms was a symbol of various beliefs. In pre-Christian times it was a pagan religious symbol throughout Europe and western Asia. In ancient times, the effigy of a man hanging on a cross was set up in the fields to protect the crops. It often appeared in conjunction with the female-genital circle or oval, to signify the sacred marriage, as in Egyptian amulet Nefer with male cross and female orb, considered as an amulet of blessedness, a charm of sexual harmony.


title: Cross

text: The word "cross" is recorded in 11th-century Old English as "cros", exclusively for the instrument of Christ's crucifixion, replacing the native Old English word "rood". The word's history is complicated; it appears to have entered English from Old Irish, possibly via Old Norse, ultimately from the Latin (or its accusative and its genitive ), "stake, cross". The English verb "to cross" arises from the noun , first in the sense "to make the sign of the cross"; the generic meaning "to intersect" develops in the 15th century. The Latin word was, however, influenced by popular etymology by a native Germanic word reconstructed as *"krukjo" (English "crook", Old English , Old Norse , Old High German ). This word, by conflation with Latin , gave rise to Old French (modern French ), the term for a shepherd's crook, adopted in English as "crosier".


title: Cross

text: Latin referred to the gibbet where criminals were executed, a stake or pole, with or without , on which the condemned were impaled or hanged, but more particularly a cross or the pole of a carriage. The derived verb means "to put to death on the cross" or, more frequently, "to put to the rack, to torture, torment", especially in reference to mental troubles.


title: Cross

text: The field of etymology is of no help in any effort to trace a supposed original meaning of "crux". A "crux" can be of various shapes: from a single beam used for impaling or suspending () to the various composite kinds of cross () made from more beams than one. The latter shapes include not only the traditional †-shaped cross (the ), but also the T-shaped cross (the or tau cross), which the descriptions in antiquity of the execution cross indicate as the normal form in use at that time, and the X-shaped cross (the "crux decussata" or saltire).


title: Cross

text: The Greek equivalent of Latin "crux" "stake, gibbet" is , found in texts of four centuries or more before the gospels and always in the plural number to indicate a stake or pole. From the first century BC, it is used to indicate an instrument used in executions. The Greek word is used in descriptions in antiquity of the execution cross, which indicate that its normal shape was similar to the Greek letter tau (Τ).


title: Cross

text: Due to the simplicity of the design (two intersecting lines), cross-shaped incisions make their appearance from deep prehistory; as petroglyphs in European cult caves, dating back to the beginning of the Upper Paleolithic, and throughout prehistory to the Iron Age.


title: Cross

text: Also of prehistoric age are numerous variants of the simple cross mark, including the "crux gammata" with curving or angular lines, and the Egyptian "crux ansata" with a loop.


title: Cross

text: Speculation has associated the cross symbol – even in the prehistoric period – with astronomical or cosmological symbology involving


title: Cross

text: "four elements" (Chevalier, 1997) or the cardinal points, or the unity of a vertical axis mundi or celestial pole with the horizontal world (Koch, 1955). Speculation of this kind became especially popular in the mid- to late-19th century in the context of comparative mythology seeking to tie Christian mythology to ancient cosmological myths. Influential works in this vein included


title: Cross

text: In the European Bronze Age the cross symbol appeared to carry a religious meaning, perhaps as a symbol of consecration, especially pertaining to burial.


title: Cross

text: The cross sign occurs trivially in tally marks, and develops into a number symbol independently in the Roman numerals (X "ten"), the Chinese rod numerals (十 "ten") and the Brahmi numerals ("four", whence the numeral 4).


title: Cross

text: In the Phoenician alphabet and derived scripts, the cross symbol represented the phoneme /t/, i.e. the letter taw, which is the historical predecessor of Latin T. The letter name "taw" means "mark", presumably continuing the Egyptian hieroglyph "two crossed sticks" (Gardiner Z9).


title: Cross

text: According to W. E. Vine's "Expository Dictionary of New Testament Words", worshippers of Tammuz in Chaldea and thereabouts used the cross as symbol of that god.


title: Cross

text: The shape of the cross ("crux", "stauros" "stake, gibbet"), as represented by the letter T, came to be used as a "seal" or symbol of Early Christianity by the 2nd century. Clement of Alexandria in the early 3rd century calls it ("the Lord's sign") he repeats the idea, current as early as the Epistle of Barnabas, that the number 318 (in Greek numerals, ΤΙΗ) in Genesis 14:14 was a foreshadowing (a "type") of the cross (the letter Tau) and of Jesus (the letters Iota Eta). Clement's contemporary Tertullian rejects the accusation that Christians are "crucis religiosi" (i.e. "adorers of the gibbet"), and returns the accusation by likening the worship of pagan idols to the worship of poles or stakes.


title: Cross

text: In his book "De Corona", written in 204, Tertullian tells how it was already a tradition for Christians to trace repeatedly on their foreheads the sign of the cross.


title: Cross

text: While early Christians used the T-shape to represent the cross in writing and gesture, the use of the Greek cross and Latin cross, i.e. crosses with intersecting beams, appears in Christian art towards the end of Late Antiquity. An early example of the cruciform halo, used to identify Christ in paintings, is found in the "Miracles of the Loaves and Fishes" mosaic of Sant'Apollinare Nuovo, Ravenna (6th century).


title: Cross

text: The Patriarchal cross, a Latin cross with an additional horizontal bar, first appears in the 10th century.


title: Cross

text: A wide variation of cross symbols is introduced for the purposes of heraldry beginning in the age of the Crusades.


title: Cross

text: Egyptian hieroglyphs involving cross shapes include "ankh" "life", "ndj" "protect" and "nfr" "good; pleasant, beautiful".


title: Cross

text: Sumerian cuneiform had a simple cross-shaped character, consisting of a horizontal and a vertical wedge (𒈦), read as "maš" "tax, yield, interest"; the superposition of two diagonal wedges results in a decussate cross (𒉽), read as "pap" "first, pre-eminent" (the superposition of these two types of crosses results in the eight-pointed star used as the sign for "sky" or "deity" (𒀭), DINGIR).


title: Cross

text: The cuneiform script has other, more complex, cruciform characters, consisting of an arrangement of boxes or the fourfold arrangement of other characters, including the archaic cuneiform characters LAK-210, LAK-276, LAK-278, LAK-617 and the classical sign EZEN (𒂡).


title: Cross

text: Phoenician "tāw" is still cross-shaped in Paleo-Hebrew alphabet and in some Old Italic scripts (Raetic and Lepontic), and its descendant T becomes again cross-shaped in the Latin minuscule t.


title: Cross

text: The plus sign (+) is derived from Latin t via a simplification of a ligature for "et" "and" (introduced by Johannes Widmann in the late 15th century).


title: Cross

text: Egyptian hieroglyphs with cross-shapes include Gardiner Z9 – Z11 ("crossed sticks", "crossed planks").


title: Cross

text: Other, unrelated cross-shaped letters include Brahmi "ka" (predecessor of the Devanagari letter क) and Old Turkic (Orkhon) "d²" and Old Hungarian "b", and Katakana ナ "na" and メ"me".


title: Cross

text: The multiplication sign (×), often attributed to William Oughtred (who first used it in an appendix to the 1618 edition of John Napier's "Descriptio") apparently had been in occasional use since the mid 16th century.


title: Cross

text: The Miscellaneous Symbols block (U+2626 to U+262F) adds three specific Christian cross variants, viz. the Patriarchal cross (☦), Cross of Lorraine (☨) and Cross potent (☩, mistakenly labeled a "Cross of Jerusalem").


title: Cross

text: The following is a list of cross symbols, "except" for variants of the Christian cross and Heraldic crosses, for which see the dedicated lists at Christian cross variants and Crosses in heraldry, respectively.


title: Cross

text: Cross shapes are made by a variety of physical gestures. Crossing the fingers of one hand is a common invocation of the symbol. The sign of the cross associated with Christian genuflection is made with one hand: in Eastern Orthodox tradition the sequence is head-heart-right shoulder-left shoulder, while in Oriental Orthodox, Catholic and Anglican tradition the sequence is head-heart-left-right.


title: Cross

text: Crossing the index fingers of both hands represents and a charm against evil in European folklore. Other gestures involving more than one hand include the "cross my heart" movement associated with making a promise and the Tau shape of the referee's "time out" hand signal.


title: Zig Ziglar

text: Hilary Hinton Ziglar (November 6, 1926 – November 28, 2012) was an American author, salesman, and motivational speaker.


title: Zig Ziglar

text: Zig Ziglar was born prematurely in Coffee County, Alabama, to John Silas Ziglar and Lila Wescott Ziglar. He was the tenth of 12 children, and the youngest boy.


title: Zig Ziglar

text: In 1931, when Ziglar was five years old, his father took a management position at a Mississippi farm, and his family moved to Yazoo City, Mississippi, where he spent most of his early childhood. The next year, his father died of a stroke, and his younger sister died two days later.


title: Zig Ziglar

text: Between 1943 and 1945, he participated in the Navy V-12 Navy College Training Program at the University of South Carolina in Columbia, South Carolina.


title: Zig Ziglar

text: Ziglar dropped out of college in 1947 and moved to Lancaster, South Carolina, where he took up a job as a salesman with the WearEver Cookware company. Ziglar was promoted to field manager and eventually divisional supervisor in 1950.


title: Zig Ziglar

text: While working at the company, Ziglar became interested in self-help and motivational speaking and began giving speeches of his own. With Richard "Dick" Gardner and Hal Krause, Ziglar was a charter member in the establishment of American Salesmasters in 1963. The company's objective was to raise the image of salespeople in America by providing seminars. They began with cities across the South and Midwest (Memphis, Atlanta, Kansas City, St. Louis, Chicago, Denver, etc.), featuring speakers such as Ziglar, Norman Vincent Peale, Ken McFarland, Cavett Robert, Bill Gove, Maxwell Maltz, and Red Motley. They booked an auditorium, put together a slate of speakers and contacted local businesses to sell tickets. Audiences included insurance agents, car salesmen, financial advisors, entrepreneurs, small business owners and curiosity seekers.


title: Zig Ziglar

text: Ziglar went on to speak extensively for audiences of the National Association of Sales Education (NASE), founded by Dick Gardner in 1965, and also became a major sales trainer for Mary Kay Cosmetics. In 1968, he became a vice president and training director for the Automotive Performance Company and moved to Dallas, Texas. The company went bankrupt two years later.


title: Zig Ziglar

text: He later founded the Zigmanship Institute, later known as Ziglar, Inc. Subsequently, Ziglar spoke extensively at seminars for motivational speaker Peter Lowe and eventually signed an exclusive agreement to support Peter Lowe events.


title: Zig Ziglar

text: In 2007, a fall down a flight of stairs left him with short-term memory problems. Nonetheless, Ziglar continued taking part in motivational seminars until he retired in 2010.


title: Zig Ziglar

text: Ziglar met his wife, Jean, in 1944, in Jackson, Mississippi. He was 17 and she was 16; they married in late 1946. They had four children: Suzan, Tom, Cindy, and Julie.


title: Zig Ziglar

text: Ziglar, a Baptist, integrated Christianity into his motivational work. He was also a Republican who endorsed former Governor of Arkansas Mike Huckabee for his party's presidential nomination in 2008.


title: Juno (spacecraft)

text: Juno is a NASA space probe orbiting the planet Jupiter. It was built by Lockheed Martin and is operated by NASA Jet Propulsion Laboratory. The spacecraft was launched from Cape Canaveral Air Force Station on August 5, 2011 UTC, as part of the New Frontiers program. "Juno" entered a polar orbit of Jupiter on July 5, 2016, UTC, to begin a scientific investigation of the planet. After completing its mission, "Juno" will be intentionally deorbited into Jupiter's atmosphere.


title: Juno (spacecraft)

text: "Juno" mission is to measure Jupiter's composition, gravitational field, magnetic field, and polar magnetosphere. It will also search for clues about how the planet formed, including whether it has a rocky core, the amount of water present within the deep atmosphere, mass distribution, and its deep winds, which can reach speeds up to .


title: Juno (spacecraft)

text: "Juno" is the second spacecraft to orbit Jupiter, after the nuclear powered "Galileo" orbiter, which orbited from 1995 to 2003. Unlike all earlier spacecraft sent to the outer planets, "Juno" is powered by solar panels, commonly used by satellites orbiting Earth and working in the inner Solar System, whereas radioisotope thermoelectric generators are commonly used for missions to the outer Solar System and beyond. For "Juno", however, the three largest solar panel wings ever deployed on a planetary probe play an integral role in stabilizing the spacecraft as well as generating power.


title: Juno (spacecraft)

text: A NASA compilation of mission names and acronyms referred to the mission by the backronym "Jupiter Near-polar Orbiter". However the project itself has consistently described it as a name with mythological associations and not an acronym. The spacecraft's current name is in reference to the Roman goddess Juno. "Juno" is sometimes called the "New Frontiers 2" as the second mission in the New Frontiers program, but is not to be confused with New Horizons 2, a proposed but unselected New Frontiers mission.


title: Juno (spacecraft)

text: "Juno" was selected on June 9, 2005, as the next New Frontiers mission after "New Horizons". The desire for a Jupiter probe was strong in the years prior to this, but there had not been any approved missions. The Discovery Program had passed over the somewhat similar but more limited Interior Structure and Internal Dynamical Evolution of Jupiter (INSIDE Jupiter) proposal, and the turn-of-the-century era Europa Orbiter was canceled in 2002. The flagship-level Europa Jupiter System Mission was in the works in the early 2000s, but funding issues resulted in it evolving into ESA's Jupiter Icy Moons Explorer.


title: Juno (spacecraft)

text: "Juno" completed a five-year cruise to Jupiter, arriving on July 5, 2016. The spacecraft traveled a total distance of roughly to reach Jupiter. The spacecraft was designed to orbit Jupiter 37 times over the course of its mission. This was originally planned to take 20 months.


title: Juno (spacecraft)

text: "Juno" trajectory used a gravity assist speed boost from Earth, accomplished by an Earth flyby in October 2013, two years after its launch on August 5, 2011. The spacecraft performed an orbit insertion burn to slow it enough to allow capture. It was expected to make three 53-day orbits before performing another burn on December 11, 2016, that would bring it into a 14-day polar orbit called the Science Orbit. Because of a suspected problem in "Juno" main engine, the burn scheduled on December 11, 2016, was cancelled and "Juno" remained in its 53-day orbit until the first Ganymede encounter of its Extended Mission. This extended mission began with a flyby of Ganymede on June 7, 2021. Subsequent flybys of Europa and then Io will further decrease the orbital period to 33 days by February 2024.


title: Juno (spacecraft)

text: During the science mission, infrared and microwave instruments will measure the thermal radiation emanating from deep within Jupiter's atmosphere. These observations will complement previous studies of its composition by assessing the abundance and distribution of water, and therefore oxygen. This data will provide insight into Jupiter's origins. "Juno" will also investigate the convection that drives natural circulation patterns in Jupiter's atmosphere. Other instruments aboard "Juno" will gather data about its gravitational field and polar magnetosphere. The "Juno" mission was planned to conclude in February 2018 after completing 37 orbits of Jupiter, but now has been commissioned through 2025 to do a further 42 additional orbits of Jupiter as well as close flybys of Ganymede, Europa and Io. The probe was then intended to be deorbited and burnt up in Jupiter's outer atmosphere to avoid any possibility of impact and biological contamination of one of its moons.


title: Juno (spacecraft)

text: "Juno" was launched atop the Atlas V at Cape Canaveral Air Force Station (CCAFS), Florida on August 5, 2011, 16:25:00 UTC. The Atlas V (AV-029) used a Russian-built RD-180 main engine, powered by kerosene and liquid oxygen. At ignition it underwent checkout 3.8 seconds prior to the ignition of five strap-on solid rocket boosters (SRBs). Following the SRB burnout, about 93 seconds into the flight, two of the spent boosters fell away from the vehicle, followed 1.5 seconds later by the remaining three. When heating levels had dropped below predetermined limits, the payload fairing that protected "Juno" during launch and transit through the thickest part of the atmosphere separated, about 3 minutes 24 seconds into the flight. The Atlas V main engine cut off 4 minutes 26 seconds after liftoff. Sixteen seconds later, the Centaur second stage ignited, and it burned for about 6 minutes, putting the satellite into an initial parking orbit. The vehicle coasted for about 30 minutes, and then the Centaur was reignited for a second firing of 9 minutes, placing the spacecraft on an Earth escape trajectory in a heliocentric orbit.


title: Juno (spacecraft)

text: Prior to separation, the Centaur stage used onboard reaction engines to spin "Juno" up to 1.4 r.p.m. About 54 minutes after launch, the spacecraft separated from the Centaur and began to extend its solar panels. Following the full deployment and locking of the solar panels, "Juno" batteries began to recharge. Deployment of the solar panels reduced "Juno" spin rate by two-thirds. The probe is spun to ensure stability during the voyage and so that all instruments on the probe are able to observe Jupiter.


title: Juno (spacecraft)

text: The voyage to Jupiter took five years, and included two orbital maneuvers in August and September 2012 and a flyby of the Earth on October 9, 2013. When it reached the Jovian system, "Juno" had traveled approximately .


title: Juno (spacecraft)

text: After traveling for about a year in an elliptical heliocentric orbit, "Juno" fired its engine twice in 2012 near aphelion (beyond the orbit of Mars) to change its orbit and return to pass by the Earth at a distance of 559 kilometers in October 2013. It used Earth's gravity to help slingshot itself toward the Jovian system in a maneuver called a gravity assist. The spacecraft received a boost in speed of more than , and it was set on a course to Jupiter. The flyby was also used as a rehearsal for the "Juno" science team to test some instruments and practice certain procedures before the arrival at Jupiter.


title: Juno (spacecraft)

text: Jupiter's gravity accelerated the approaching spacecraft to around . On July 5, 2016, between 03:18 and 03:53 UTC Earth-received time, an insertion burn lasting 2,102 seconds decelerated "Juno" by and changed its trajectory from a hyperbolic flyby to an elliptical, polar orbit with a period of about 53.5 days. The spacecraft successfully entered Jovian orbit on July 5, 2016, at 03:53 UTC.


title: Juno (spacecraft)

text: "Juno" highly elliptical initial polar orbit takes it within of the planet and out to , far beyond Callisto's orbit. An eccentricity-reducing burn, called the Period Reduction Maneuver, was planned that would drop the probe into a much shorter 14 day science orbit. Originally, "Juno" was expected to complete 37 orbits over 20 months before the end of its mission. Due to problems with helium valves that are important during main engine burns, mission managers announced on February 17, 2017, that "Juno" would remain in its original 53-day orbit, since the chance of an engine misfire putting the spacecraft into a bad orbit was too high. "Juno" completed only 12 science orbits before the end of its budgeted mission plan, ending July 2018. In June 2018, NASA extended the mission through July 2021, as described below.


title: Juno (spacecraft)

text: The orbits were carefully planned in order to minimize contact with Jupiter's dense radiation belts, which can damage spacecraft electronics and solar panels, by exploiting a gap in the radiation envelope near the planet, passing through a region of minimal radiation. The "Juno Radiation Vault", with 1-centimeter-thick titanium walls, also aids in protecting "Juno" electronics. Despite the intense radiation, JunoCam and the Jovian Infrared Auroral Mapper (JIRAM) are expected to endure at least eight orbits, while the Microwave Radiometer (MWR) should endure at least eleven orbits. "Juno" will receive much lower levels of radiation in its polar orbit than the "Galileo" orbiter received in its equatorial orbit. "Galileo" subsystems were damaged by radiation during its mission, including an LED in its data recording system.


title: Juno (spacecraft)

text: The spacecraft completed its first flyby of Jupiter (perijove 1) on August 26, 2016, and captured the first images of the planet's north pole.


title: Juno (spacecraft)

text: On October 14, 2016, days prior to perijove 2 and the planned Period Reduction Maneuver, telemetry showed that some of "Juno" helium valves were not opening properly. On October 18, 2016, some 13 hours before its second close approach to Jupiter, "Juno" entered into safe mode, an operational mode engaged when its onboard computer encounters unexpected conditions. The spacecraft powered down all non-critical systems and reoriented itself to face the Sun to gather the most power. Due to this, no science operations were conducted during perijove 2.


title: Juno (spacecraft)

text: On December 11, 2016, the spacecraft completed perijove 3, with all but one instrument operating and returning data. One instrument, JIRAM, was off pending a flight software update. Perijove 4 occurred on February 2, 2017, with all instruments operating. Perijove 5 occurred on March 27, 2017. Perijove 6 took place on May 19, 2017.


title: Juno (spacecraft)

text: Although the mission's lifetime is inherently limited by radiation exposure, almost all of this dose was planned to be acquired during the perijoves. , the 53.4 day orbit was planned to be maintained through July 2018 for a total of twelve science-gathering perijoves. At the end of this prime mission, the project was planned to go through a science review process by NASA's Planetary Science Division to determine if it will receive funding for an extended mission.


title: Juno (spacecraft)

text: In June 2018, NASA extended the mission operations plan to July 2021. During the mission, the spacecraft will be exposed to high levels of radiation from Jupiter's magnetosphere, which may cause future failure of certain instruments and risk collision with Jupiter's moons.


title: Juno (spacecraft)

text: In January 2021, NASA extended the mission operations to September 2025. In this phase "Juno" began to examine Jupiter's inner moons, Ganymede, Europa and Io. A flyby of Ganymede occurred on June 7, 2021, 17:35 UTC, coming within , the closest any spacecraft has come to the moon since "Galileo" in 2000. A flyby of Europa took place on September 29, 2022, at a distance of . Finally, the spacecraft is scheduled to perform two flybys of Io in 2024 at a distance of . These flybys will further help with upcoming missions including NASA's Europa Clipper Mission and the European Space Agency's JUICE (JUpiter ICy moons Explorer), as well as the proposed Io Volcano Observer.


title: Juno (spacecraft)

text: NASA originally planned to deorbit the spacecraft into the atmosphere of Jupiter after completing 32 orbits of Jupiter, but has since extended the mission to September 2025. The controlled deorbit is intended to eliminate space debris and risks of contamination in accordance with NASA's planetary protection guidelines.


title: Juno (spacecraft)

text: Scott Bolton of the Southwest Research Institute in San Antonio, Texas is the principal investigator and is responsible for all aspects of the mission. The Jet Propulsion Laboratory in California manages the mission and the Lockheed Martin Corporation was responsible for the spacecraft development and construction. The mission is being carried out with the participation of several institutional partners. Coinvestigators include Toby Owen of the University of Hawaii, Andrew Ingersoll of California Institute of Technology, Frances Bagenal of the University of Colorado at Boulder, and Candy Hansen of the Planetary Science Institute. Jack Connerney of the Goddard Space Flight Center served as instrument lead.


title: Juno (spacecraft)

text: "Juno" was originally proposed at a cost of approximately US$700 million (fiscal year 2003) for a launch in June 2009 (equivalent to US$ million in ). NASA budgetary restrictions resulted in postponement until August 2011, and a launch on board an Atlas V rocket in the 551 configuration. the mission was projected to cost US$1.46 billion for operations and data analysis through 2022.


title: Juno (spacecraft)

text: The "Juno" mission's scientific objectives will be achieved with a payload of nine instruments on board the spacecraft:


title: Juno (spacecraft)

text: The microwave radiometer comprises six antennas mounted on two of the sides of the body of the probe. They will perform measurements of electromagnetic waves on frequencies in the microwave range: 600 MHz, 1.2, 2.4, 4.8, 9.6 and 22 GHz, the only microwave frequencies which are able to pass through the thick Jovian atmosphere. The radiometer will measure the abundance of water and ammonia in the deep layers of the atmosphere up to pressure or deep. The combination of different wavelengths and the emission angle should make it possible to obtain a temperature profile at various levels of the atmosphere. The data collected will determine how deep the atmospheric circulation is. The MWR is designed to function through orbit 11 of Jupiter.