Friday, June 4, 2010

Falcon 9 - Off We Go, Into the Wild Blue Yonder

Click here for the webcam.

Falcon 9 has a 4-hr launch window from 11 am - 3 pm EDT today for its initial flight. Atlas required 14 failed launch attempts before succeeding. Behold commercial spaceflight's new workhorse:


Falcon 9 is a spaceflight launch system that uses rockets designed and manufactured by SpaceX. Both stages of the two-stage-to-orbit vehicles, which use liquid oxygen (LOX) and rocket-grade kerosene (RP-1) propellants, are intended to be reusable.[4] Multiple variants are planned with payloads of between 10,450 kg and 26,610 kg to low Earth orbit, and between 4,540 kg and 15,010 kg to geostationary transfer orbit, placing the Falcon 9 design in the medium-lift to heavy-lift range of launch systems.

As of February 23, 2010, the first Falcon 9 is upright on its launch pad at Launch Complex 40 in preparation for its maiden launch. Currently the maiden flight is tentatively scheduled for launch on June 4, 2010,[5] delayed from Nov. 29, Feb. 9, March 3, March 8, May 16, May 27 and June 2.[2] SpaceX has repeatedly stated that all launch dates are subject to change as they work through various issues and the scheduled launch date should be treated as the earliest possible date, not the actual launch.[6] However the June 4 date is now subject only to weather conditions as the testing of the FTS has been completed and is nominal. Falcon 9 will be the first new rocket to launch from Cape Canaveral since 2002.[7]

The Falcon 9 will be the launch vehicle for the SpaceX Dragon spacecraft. The Falcon 9 and Dragon combination won a Commercial Resupply Services (CRS) contract from NASA to resupply the International Space Station under the Commercial Orbital Transportation Services (COTS) program. Bigelow Aerospace is also considering the Falcon 9 for their Orion Lite manned spacecraft.

Contents

Design

Falcon 9 (left) and Falcon 9 Heavy (right)

The base Falcon 9 is a two stage, LOX/RP-1 powered launch vehicle. Its first stage is powered by nine SpaceX Merlin 1C rocket engines with 125,000 lbf sea-level thrust per engine for a total thrust on liftoff of approximately 4.9 million N (1.1 million lbf).[8] The Falcon 9 first stage uses a pyrophoric mixture of triethylaluminum-triethylborane (TEA-TEB) as a first-stage ignitor. [9]

The Falcon 9 Heavy consists of a standard Falcon 9 with two additional Falcon 9 first stages acting as liquid strap-on boosters,[10] which is conceptually similar to EELV launchers Delta IV Heavy and the future Atlas V HLV, and also to the Russian Angara carrier rocket.

The upper stage is powered by a single Merlin engine modified for vacuum operation with an expansion ratio of 117:1 and a nominal burn time of 345 seconds. For added reliability of restart, the engine has dual redundant pyrophoric igniters (TEA-TEB).[8] SpaceX has expressed hopes that both stages will eventually be reusable.[4]

The interstage, which connects the upper and lower stage for Falcon 9, is a carbon fiber aluminum core composite structure. Stage separation occurs via reusable separation collets and a pneumatic pusher system. The Falcon 9 tank walls and domes are made from aluminum lithium alloy. SpaceX uses an all friction stir welded tank, the highest strength and most reliable welding technique available. The second stage tank of Falcon 9 is simply a shorter version of the first stage tank and uses most of the same tooling, material and manufacturing techniques. This results in significant cost savings in vehicle production.[8]

As with the company's smaller Falcon 1 vehicle, Falcon 9's launch sequence includes a hold-down feature that allows full engine ignition and systems check before liftoff. After first stage engine start, the Falcon is held down and not released for flight until all propulsion and vehicle systems are confirmed to be operating normally. An automatic safe shut-down and unloading of propellant occurs if any off nominal conditions are detected.[8]

Falcon 9 will have triple redundant flight computers and inertial navigation, with a GPS overlay for additional orbit insertion accuracy.[8]

The Falcon 9 will launch from Cape Canaveral Air Force Station Space Launch Complex 40.

Launcher versions

Version Falcon 9 Falcon 9 Heavy
Stage 0 — 2 boosters with 9 × Merlins 1C each
Stage 1 9 × Merlin 1C 9 × Merlin 1C
Stage 2 1 × Merlin 1C 1 × Merlin 1C
Height
(max; m)		 50 or 54 (large fairing) 50 or 54 (large fairing)
Diameter
(m)		 3.6 3.6 or 5.2 (large fairing)[10]
Initial thrust
(kN)		 4,400 12,258
Takeoff weight
(tonnes)		 325 885
Fairing diameter
(Inner; m)		 4.6 4.6
Payload
(LEO; kg)		 8,560 (polar orbit from Kwajalein) or 10,450 (launch at Cape Canaveral) 29,610
Payload
(GTO; kg)		 4,680 (launch at Kwajalein) or 4,540 (launch at Cape Canaveral) 15,010
Price
(Mil. USD)		 35 to LEO; 35 to 55 (according to Satellite Mass) to GEO 90 to LEO; 55 to 90 (according to Satellite Mass) to GEO
minimal Price/kg
(LEO; USD)		 3,365 3,273
minimal Price/kg
(GTO; USD)		 between 10,000 and 11,000 (according to Satellite Mass) between 7,826 and 10,000 (according to Satellite Mass)
Success ratio
(successful/total)		 — —

[11] [12] [13] [14]

Initial descriptions

At an appearance in May 2004 before the U.S. Senate Committee on Commerce, Science and Transportation, Elon Musk testified, "Long term plans call for development of a heavy lift product and even a super-heavy, if there is customer demand. [...] Ultimately, I believe $500 per pound [of payload delivered to orbit] or less is very achievable."[15]

SpaceX formally announced the Falcon 9 on 2005-09-08, describing it as being a "fully reusable heavy lift launch vehicle."[16] A Falcon 9 medium was described as being capable of launching approximately 21,000 lb (9,500 kg) to low Earth orbit, priced at $27 million per flight ($1286/lb).

Production and testing

On April 12, 2007 SpaceX announced it had completed the primary structure for its first Falcon 9 first-stage tank.[17] The tank was shipped to a SpaceX test facility in Texas for first-stage static firing validation. The first multi-engine test (with two engines connected to the first stage, firing simultaneously) was successfully completed in January 2008. On March 8, 2008, three Merlin 1C engines were fired simultaneously for the first time. The next test took place on May 29, 2008, and saw five engines firing together. The first nine-engine firing tests were conducted on July 31 and August 1, 2008; both were successful.[18][19][20] On November 22, 2008 the full Falcon 9 complement of nine engines was test fired for a full mission length (178 seconds) of the first stage.[21]

In February 2008, the plan was for the first Falcon 9/Dragon COTS Demo flight to be delayed by six months to late in the first quarter of 2009, due to the immense amount of development and regulatory work required. According to Elon Musk, the complexity of the development work and the regulatory requirements for launching from Cape Canaveral have contributed to the delay.[22]

In January 2009, Falcon 9 was first raised to the vertical position at Space Launch Complex 40 in Cape Canaveral. In October 2009, the first flight-ready first stage had a successful all-engine test fire at the company's test stand in McGregor, TX. The full stack had arrived at the launch site for integration at the beginning of February 2010. SpaceX scheduled a launch date of March 22, 2010, though they estimate anywhere between one and three months for integration and testing.[23]

In November 2009 Space X conducted the initial second stage test firing lasting forty seconds. This test involved a new test stand, a new flight stage, and it occurred as planned, on the first attempt without aborts or recycles.

On January 2, 2010, A successful full duration orbit insertion firing of the Falcon 9 second stage was conducted at the McGregor test site. SpaceX completed a full duration orbit insertion firing (329 seconds) of the integrated Falcon 9 second stage.

Thursday, February 25, 2010 SpaceX's Falcon 9 launch vehicle was set vertical at Space Launch Complex 40, Cape Canaveral.[24]

Tuesday, March 9th, SpaceX performed a Static Fire for the Falcon 9 launch vehicle. Some fire and smoke were seen at the base of the rocket, leading to speculation of an engine fire. However, all components checked out, but the test executed a nominal abort at T-2 seconds due to a failure in the spin-start system. This system is designed to pump high pressure helium from the launch pad into the first stage turbopumps to get them spinning in preparation for launch. Subsequent review showed that the failure point was a valve that didn't receive a command to open. As the problem was with the pad and not with the rocket itself, it didn't occur at the McGregor test site, which didn't have the same valve setup. No damage was sustained by the vehicle or the test pad and the fire and smoke were the result of normal burnoff from the liquid oxygen and fuel mix present in the system prior to launch. All vehicle systems leading up to the abort performed as expected and no additional issues were noted that needed addressing. A subsequent test on March 13 was successful in firing the nine first stage engines for 3.5 seconds.[25]

Delays since March 13th have been mainly attributed to review of the Falcon 9 flight termination system by the Air Force.

On June 1st, SpaceX announced on their update page that they had completed testing of the FTS and all results were nominal, thus they were expecting formal approval in time to launch on June 4th.

Scheduled launches

According to the USAF, which operates the Eastern Range, the first Falcon 9 launch was due to take place no earlier than November 29, 2009.[26] The launch was later moved to February 9, 2010, and eventually rescheduled for no earlier than March 2010. Launch dates were set for Feb. 9, March 3, March 8 and March 22, April 12, May 16, May 23, May 27 all rescheduled to the current launch date of June 4, 2010.[2]

(Dates provided by the SpaceX Launch Manifest are "target" dates for vehicle arrival at launch site)

  • Q4 2009 (arrival at launch site): Demonstration flight of Falcon 9 from Cape Canaveral.[27] Launch is planned on June 4, 2010.[2][28][29] Flight will carry a Dragon Spacecraft Qualification Unit as a payload.
  • 2010: Demo flight 1 of Falcon 9/Dragon for NASA Commercial Orbital Transportation Services (COTS) program.[27][30] The flight will also carry two cubesats as part of the NRO's colony program.[31] Launch was expected in Q1 2010[32]
  • 2010: Demo flight 2 of Falcon 9/Dragon for NASA COTS program,[27] 2nd stage becomes a rendezvous target for the Dragon capsule (per SpaceX manifest)[27]
  • 2010: Demo flight 3 of Falcon 9/Dragon for NASA COTS program,[27] demonstration of cargo delivery to the International Space Station (per SpaceX manifest)[27]
  • 2011: MDA Corporation payload on Falcon 9[27][30]
  • 2011: Resupply mission 1 for NASA COTS program,[27] cargo delivery to the International Space Station (per SpaceX manifest)[27]
  • 2011: Resupply mission 2 for NASA COTS program,[27] cargo delivery to the International Space Station (per SpaceX manifest)[27]
  • 2012: DragonLab Mission 1. Non-COTS private mission of the Falcon 9 and Dragon spaceship - payload not yet public (per SpaceX manifest)[27]

There are twelve total Falcon 9 resupply missions contracted to the ISS between 2010 and 2015.[27]

On March 15, 2010, SpaceX announced that they had signed an agreement to launch a Space Systems/Loral-built satellite on a Falcon 9, “as early as 2012.”[33]

Comparable rockets

These rockets are all heavy lift launch vehicles (over 14,000 kg to LEO and over 5,000 kg to GTO). Most of them use liquid oxygen with liquid hydrogen or kerosene.

See also

Notes

  1. ^ a b c d "Launch Cost Falcon 9". SpaceX. http://www.spacex.com/falcon9.php#pricing_and_performance.
  2. ^ a b c d "Detailed Mission Data - Falcon-9 ELV First Flight Demonstration". Mission Set Database. NASA GSFC. http://msdb.gsfc.nasa.gov/MissionData.php?mission=Falcon-9%20ELV%20First%20Flight%20Demonstration. Retrieved 2010-05-26.
  3. ^ SpaceX (March 10, 2009). "SpaceX Falcon 9 Upper Stage Engine Successfully Completes Full Mission Duration Firing". Press release. http://www.spacex.com/press.php?page=20090310.
  4. ^ a b Musk ambition: SpaceX aim for fully reusable Falcon 9, NASAspaceflight.com, 2009-01-12, accessed 2010-06-03. E. Musk Quotation: "By flight six we think it’s highly likely we’ll recover the first stage, and when we get it back we’ll see what survived through re-entry, and what got fried, and carry on with the process. ... That’s just to make the first stage reusable, it’ll be even harder with the second stage – which has got to have a full heatshield, it’ll have to have deorbit propulsion and communication."
  5. ^ "Next Launch". Patrick Air Force Base. http://www.patrick.af.mil/. Retrieved 31 May 2010.
  6. ^ Clark, Stephen (February 20, 2010). "Falcon 9 rocket will spend next week at launch pad". Spaceflight Now. http://www.spaceflightnow.com/falcon9/001/100220rollout/.
  7. ^ Peterson, Patrick (February 23, 2010). "SpaceX prepares to let Falcon fly from Cape". Florida Today. http://www.floridatoday.com/article/20100223/NEWS02/2230321/1006/news01/Falcon+9+prepares+to+soar.
  8. ^ a b c d e "Falcon 9 Overview". SpaceX. 08 May 2010. http://www.spacex.com/falcon9.php.
  9. ^ Mission Status Center, June 2, 2010, 1905 GMT, [SpaceflightNow]], accessed 2010-06-02, Quotation: "The flanges will link the rocket with ground storage tanks containing liquid oxygen, kerosene fuel, helium, gaserous nitrogen and the first stage ignitor source called triethylaluminum-triethylborane, better known as TEA-TAB."
  10. ^ a b SpaceX hopes to supply ISS with new Falcon 9 heavy launcher, Flight International, September 13, 2005, accessed 2010-06-03.
  11. ^ "SpaceX Falcon Data Sheet". Space Launch Report. 5 July 2007. http://www.spacelaunchreport.com/falcon.html#config.
  12. ^ "Monster Progress Update (Mostly Falcon 9)". SpaceX. 17 August 2007. http://www.spacex.com/updates_archive.php?page=081707.
  13. ^ "Falcon 1 Overview". SpaceX. 28 September 2007. http://www.spacex.com/falcon1.php.
  14. ^ "Falcon 9 Heavy Overview". SpaceX. http://www.spacex.com/falcon9_heavy.php. Retrieved 28 May 2010.
  15. ^ Testimony of Elon Musk (May 5, 2004). "Space Shuttle and the Future of Space Launch Vehicles". U.S. Senate. http://www.spaceref.com/news/viewsr.html?pid=12774.
  16. ^ SpaceX (2005-09-08). "SpaceX Announces the Falcon 9 Fully Reusable Heavy Lift Launch Vehicle". Press release. http://www.spacex.com/press.php?page=18.
  17. ^ Space Exploration Technologies Corp. (2007-04-11). "SpaceX Completes Primary Structure of the Falcon 9 First Stage Tank". Press release. http://www.spacex.com/press.php?page=27.
  18. ^ "SpaceX: First nine engine firing of its Falcon 9". NASA Spaceflight. August 02, 2008. http://forum.nasaspaceflight.com/index.php?topic=13931.0.
  19. ^ SpaceX (18 January 2008). "SpaceX Conducts First Multi-Engine Firing of Falcon 9 Rocket". Press release. http://www.spacex.com/press.php?page=35.
  20. ^ SpaceX (27 March 2008). "SpaceX conducts first Three-Engine firing of Falcon 9 rocket". Press release. http://www.spacex.com/press.php?page=40.
  21. ^ SpaceX (November 23, 2008). "SpaceX successfully conducts full mission-length firing of its Falcon 9 launch vehicle". Press release. http://www.spacex.com/press.php?page=20081123.
  22. ^ Rob Coppinger (2008-02-27). "SpaceX Falcon 9 maiden flight delayed by six months to late Q1 2009". Flight Global. http://www.flightglobal.com/articles/2008/02/27/221883/spacex-falcon-9-maiden-flight-delayed-by-six-months-to-late-q1.html.
  23. ^ ""SpaceX announces Falcon 9 assembly underway at the Cape". Orlando Sentinel. 11 Feb 2010. http://blogs.orlandosentinel.com/news_space_thewritestuff/2010/02/spacex-announces-falcon-9-assembly-underway-at-the-cape.html.
  24. ^ ""Space X Website". http://www.spacex.com/updates.php.
  25. ^ ""universe today". http://www.universetoday.com/2010/03/13/successful-engine-test-firing-for-spacex-inaugural-falcon-9/.
  26. ^ "Air Force: SpaceX's Falcon 9 first launch planned for Nov 29". Orlando Sentinel. Sep 22, 2009. http://blogs.orlandosentinel.com/news_space_thewritestuff/2009/09/air-force-spacexs-falcon-9-first-launch-planned-for-nov-29-.html.
  27. ^ a b c d e f g h i j k l m "Launch Manifest". SpaceX. 2009. http://spacex.com/launch_manifest.php. Retrieved 2010-03-01.
  28. ^ Clark, Stephen (February 20, 2010). "Falcon 9 rocket will spend next week at launch pad". spaceflightNow.com. http://www.spaceflightnow.com/falcon9/001/100220rollout/. Retrieved February 21, 2010.
  29. ^ "Elon Musk: SpaceX Falcon 9’s maiden launch just “months” away". Orlando Sentinel. 2009-01-05. http://blogs.orlandosentinel.com/news_space_thewritestuff/2010/01/elon-musk-spacex-falcon-9s-maiden-launch-just-months-away.html. Retrieved 2010-01-06.
  30. ^ a b NASA, SpaceX (2006) (PDF). Space Act Agreement Between National Aeronautics And Space Administration And Space Explorations Technologies Corp. For Commercial Orbital Transport Services Demonstration (COTS). http://www.nasa.gov/centers/johnson/pdf/162330main_SPACE_ACT_AGREEMENT_FOR_COTS.pdf. Retrieved 2007-11-22.
  31. ^ "NRO Taps Boeing for Next Batch of Cubesats". Space News. 2010-04-08. http://www.spacenews.com/military/100408-nro-taps-boeing-next-cubesats.html. Retrieved 2010-04-12.
  32. ^ "COTS Status Update & Crew Capabilities" (PDF). SpaceX. 2009-06-17. http://www.spacex.com/20090617_Elon_Musk_Augustine_Commission.pdf. Retrieved 2009-09-21.
  33. ^ http://www.spacex.com/press.php?page=20100315

External links
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Wednesday, June 2, 2010

Billions of Galaxies Billions of Years Ago

Thousands of galaxies crowd into this Herschel image of the distant universe. Each dot is an entire galaxy containing billions of stars. This image image was taken in a region of space called the Lockman hole, which allows a clear line of sight out into the distant Universe. This "hole" is located in the familiar northern constellation of Ursa Major, The Great Bear. Credit: ESA & SPIRE Consortium & HerMES consortia

From this Space.com article.

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Monday, May 31, 2010

James Webb in NYC


A life size model of the 80' x 40' x 40' James Webb Space Telescope will be on display in New York City's Battery park beginning tomorrow June 1 through Sunday June 6.

Click here for more details.
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Sunday, May 30, 2010

Évariste Galois, forever young


Évariste Galois (October 25, 1811 – May 31, 1832) was a French mathematician born in Bourg-la-Reine. While still in his teens, he was able to determine a necessary and sufficient condition for a polynomial to be solvable by radicals, thereby solving a long-standing problem. His work laid the foundations for Galois theory, a major branch of abstract algebra, and the subfield of Galois connections. He was the first to use the word "group" (French: groupe) as a technical term in mathematics to represent a group of permutations. A radical Republican during the monarchy of Louis Philippe in France, he died from wounds suffered in a duel under shadowy circumstances[1] at the age of twenty.

Contents

Life

Early life

Galois was born on October 25, 1811, to Nicolas-Gabriel Galois and Adélaïde-Marie (born Demante). His father was a Republican and was head of Bourg-la-Reine's liberal party. He became mayor of the village after Louis XVIII returned to the throne in 1814. His mother, the daughter of a jurist, was a fluent reader of Latin and classical literature and she was responsible for her son's education for his first twelve years. At the age of 10, Galois was offered a place at the college of Reims, but his mother preferred to keep him at home.

In October 1823, he entered the Lycée Louis-le-Grand, and despite some turmoil in the school at the beginning of the term (where about a hundred students were expelled), Galois managed to perform well for the first two years, obtaining the first prize in Latin. He soon became bored with his studies, and at the age of 14, he began to take a serious interest in mathematics.

He found a copy of Adrien Marie Legendre's Éléments de Géométrie, which it is said that he read "like a novel" and mastered at the first reading. At 15, he was reading the original papers of Joseph Louis Lagrange, such as the landmark Réflexions sur la résolution algébrique des équations which likely motivated his later work on equation theory, and Leçons sur le calcul des fonctions, work intended for professional mathematicians, yet his classwork remained uninspired, and his teachers accused him of affecting ambition and originality in a negative way.[2]

Budding mathematician

In 1828, he attempted the entrance exam to École Polytechnique, the most prestigious institution for mathematics in France at the time, without the usual preparation in mathematics, and failed for lack of explanations on the oral examination. In that same year, he entered the École Normale (then known as l'École préparatoire), a far inferior institution for mathematical studies at that time, where he found some professors sympathetic to him.

In the following year, Galois' first paper, on continued fractions,[3] was published. It was at around the same time that he began making fundamental discoveries in the theory of polynomial equations. He submitted two papers on this topic to the Academy of Sciences. Augustin Louis Cauchy refereed these papers, but refused to accept them for publication for reasons that still remain unclear. However, in spite of many claims to the contrary, it appears that Cauchy recognized the importance of Galois' work, and that he merely suggested combining the two papers into one in order to enter it in the competition for the Academy's Grand Prize in Mathematics. Cauchy, a highly eminent mathematician of the time considered Galois' work to be a likely winner.[4]

On July 28, 1829, Galois' father committed suicide after a bitter political dispute with the village priest. A couple of days later, Galois took his second, and final attempt at entering the Polytechnique, and failed yet again. It is undisputed that Galois was more than qualified; however, accounts differ on why he failed. The legend holds that he thought the exercise proposed to him by the examiner to be of no interest, and, in exasperation, he threw the rag used to clean up chalk marks on the blackboard at the examiner's head.[5][6] More plausible accounts state that Galois made too many logical leaps and baffled the incompetent examiner, evoking irascible rage in Galois. The recent death of his father may have also influenced his behavior.[2]

Having been denied admission to the Polytechnique, Galois took the Baccalaureate examinations in order to enter the Ecole Normale. He passed, receiving his degree on December 29, 1829. His examiner in mathematics reported, "This pupil is sometimes obscure in expressing his ideas, but he is intelligent and shows a remarkable spirit of research."

His memoir on equation theory would be submitted several times, but was never published in his lifetime due to various events. As noted before, his first attempt was refused by Cauchy, but he tried again in February 1830 after following Cauchy's suggestions and submitted it to the Academy's secretary Fourier, to be considered for the Grand Prix of the Academy. Unfortunately, Fourier died soon after, and the memoir was lost. The prize would be awarded that year to Abel posthumously and also to Jacobi. Despite the lost memoir, Galois published three papers that year, two of which laid the foundations for Galois theory,[7][8] and the third, an important one on number theory, where the concept of a finite field was first articulated.[9]

Political firebrand

Galois lived during a time of political turmoil in France. Charles X had succeeded Louis XVIII in 1824, but in 1827 his party suffered a major electoral setback and by 1830 the opposition liberal party became the majority. Charles, faced with abdication, staged a coup d'état, and issued his notorious July Ordinances, touching off the July Revolution which ended with Louis-Philippe becoming king. While their counterparts at Polytechnique were making history in the streets during the les Trois Glorieuses, Galois and all the other students at the École Normale were locked in by the school's director. Galois was incensed and he wrote a blistering letter criticizing the director which he submitted to the Gazette des Écoles, signing the letter with his full name. Despite the fact that the Gazette's editor redacted the signature for publication, Galois was, predictably, expelled for it.[6]

Although his expulsion would have formally taken effect on January 4, 1831, Galois quit school immediately and joined the staunchly Republican artillery unit of the National Guard. These and other political affiliations continually distracted him from mathematical work. Due to controversy surrounding the unit, soon after Galois became a member, on December 31, 1830, the artillery of the National Guard was disbanded out of fear that they might destabilize the government. At around the same time, nineteen officers of Galois' former unit were arrested and charged with conspiracy to overthrow the government.

In April, the nineteen officers were acquitted of all charges, and on May 9, 1831, a banquet was celebrated in their honor, with many illustrious people present, such as Alexandre Dumas. The proceedings became more riotous, and Galois proposed a toast to King Louis-Philippe with a dagger above his cup, which was interpreted as a threat against the king's life. He was arrested the following day, but was acquitted on June 15.[6][10]

On the following Bastille Day, Galois was at the head of a protest, wearing the uniform of the disbanded artillery, and came heavily armed with several pistols, a rifle, and a dagger. For this, he was again arrested and this time sentenced to six months in prison for illegally wearing a uniform.[5] He was released on April 29, 1832. During his imprisonment, he continued developing his mathematical ideas.

Final days
The Galois memorial in the cemetery of Bourg-la-Reine. Évariste Galois was buried in a common grave and the exact location is still unknown.

Galois returned to mathematics after his expulsion from the École Normale, although he was constantly distracted by his political activities. After his expulsion became official in January 1831, he attempted to start a private class in advanced algebra which did manage to attract a fair bit of interest, but this waned, as it seemed that his political activism had priority.[2][4] Simeon Poisson asked him to submit his work on the theory of equations, which he did on January 17. Around July 4, Poisson declared Galois' work "incomprehensible", declaring that "[Galois'] argument is neither sufficiently clear nor sufficiently developed to allow us to judge its rigor"; however, the rejection report ends on an encouraging note: "We would then suggest that the author should publish the whole of his work in order to form a definitive opinion."[11] While Poisson's report was made before Galois' Bastille Day arrest, it took some time for it to reach Galois, which it finally did in October that year, while he was imprisoned. It is unsurprising, in the light of his character and situation at the time, that Galois reacted violently to the rejection letter, and he decided to forget about having the Academy publish his work, and instead publish his papers privately through his friend Auguste Chevalier. Apparently, however, Galois did not ignore Poisson's advice and began collecting all his mathematical manuscripts while he was still in prison, and continued polishing his ideas until his release on April 29, 1832.[6]

Galois' fatal duel took place on May 30. The true motives behind the duel will most likely remain forever obscure. There has been a lot of speculation, much of it spurious, as to the reasons behind it. What is known is that five days before his death, he wrote a letter to Chevalier which clearly alludes to a broken love affair.[4]

Some archival investigation on the original letters reveals that the woman he was in love with was apparently a certain Mademoiselle Stéphanie-Felicie Poterin du Motel, the daughter of the physician at the hostel where Galois remained during the final months of his life. Fragments of letters from her copied by Galois himself (with many portions either obliterated, such as her name, or deliberately omitted) are available.[12] The letters give some intimation that Mlle. du Motel had confided some of her troubles with Galois, and this might have prompted him to provoke the duel himself on her behalf. This conjecture is also supported by some of the other letters Galois later wrote to his friends the night before he died. Much more detailed speculation based on these scant historical details has been interpolated by many of Galois' biographers (most notably by Eric Temple Bell in Men of Mathematics), such as the oft-repeated conjecture that the entire incident was stage-managed by the police and royalist factions to eliminate a political enemy.[5]

As to his opponent in the duel, Alexandre Dumas names Pescheux d'Herbinville, one of the nineteen artillery officers on whose acquittal the banquet that occasioned Galois' first arrest was celebrated[10] and du Motel's fiancee.[citation needed] However, Dumas is alone in this assertion, and extant newspaper clippings from only a few days after the duel give a description of his opponent which is inconsistent with d'Herbinville, and more accurately describes one of Galois' Republican friends, most probably Ernest Duchatelet, who was also imprisoned with Galois on the same charges. Given the conflicting information available, the true identity of his killer may well be lost to history.

Whatever the reasons behind the duel, Galois was so convinced of his impending death that he stayed up all night writing letters to his Republican friends and composing what would become his mathematical testament, the famous letter to Auguste Chevalier outlining his ideas.[13] Hermann Weyl, one of the greatest mathematicians of the 20th century, said of this testament, "This letter, if judged by the novelty and profundity of ideas it contains, is perhaps the most substantial piece of writing in the whole literature of mankind." However, the legend of Galois pouring his mathematical thoughts onto paper the night before he died seems to have been exaggerated. In these final papers, he outlined the rough edges of some work he had been doing in analysis and annotated a copy of the manuscript submitted to the Academy and other papers.

Early in the morning of May 30, 1832, he was shot in the abdomen and died the following day at ten in the Cochin hospital (probably of peritonitis) after refusing the offices of a priest. He was 20 years old. His last words to his brother Alfred were:

Ne pleure pas, Alfred ! J'ai besoin de tout mon courage pour mourir à vingt ans ! (Don't cry, Alfred! I need all my courage to die at twenty.)

On June 2, Évariste Galois was buried in a common grave of the Montparnasse cemetery whose exact location is unknown.[14] In the cemetery of his native town - Bourg-la-Reine - a cenotaph in his honour was erected beside the graves of his relatives.[15]

Much of the drama surrounding the legend of his death has been attributed to one source, Eric Temple Bell's Men of Mathematics.[4]

Galois' mathematical contributions were published in full in 1843 when Liouville reviewed his manuscript and declared it sound. It was finally published in the October–November 1846 issue of the Journal de Mathématiques Pures et Appliquées.[16] The most famous contribution of this manuscript was a novel proof that there is no quintic formula, that is, that fifth and higher degree equations are not solvable by radicals. Although Abel had already proved the impossibility of a "quintic formula" by radicals in 1824 and Ruffini had published a solution in 1799 that turned out to be flawed, Galois' methods led to deeper research in what is now called Galois theory. For example, one can use it to determine, for any polynomial equation, whether it has a solution by radicals.

Contributions to Mathematics

“ Tu prieras publiquement Jacobi ou Gauss de donner leur avis, non sur la vérité, mais sur l'importance des théorèmes.

Après cela, il y aura, j'espere, des gens qui trouveront leur profit à déchiffrer tout ce gâchis.

(Ask Jacobi or Gauss publicly to give their opinion, not as to the truth, but as to the importance of these theorems. Later there will be, I hope, some people who will find it to their advantage to decipher all this mess.)

 ”

—Évariste Galois, Lettre de Galois à M. Auguste Chevalier[13]

Unsurprisingly, Galois' collected works amount to only some 60 pages, but within them are many important ideas that have had far-reaching consequences for nearly all branches of mathematics.[17] His work has been compared to that of Niels Henrik Abel, another mathematician who died at a very young age, and much of their work had significant overlap.

Algebra

While many mathematicians before Galois gave consideration to what are now known as groups, it was Galois who was the first to use the word 'group' (in French groupe) in a sense close to the technical sense that is understood today, making him among the key founders of the branch of algebra known as group theory. He developed the concept that is today known as a normal subgroup. He called the decomposition of a group into its left and right cosets a 'proper decomposition' if the left and right cosets coincide, which is what today is known as a normal subgroup.[13] He also introduced the concept of a finite field (also known as a Galois field in his honor), in essentially the same form as it is understood today.[9]

Galois Theory

Main article: Galois theory

Galois' most significant contribution to mathematics by far is his development of Galois theory. He realized that the algebraic solution to a polynomial equation is related to the structure of a group of permutations associated with the roots of the polynomial, the Galois group of the polynomial. He found that an equation could be solvable in radicals if one can find a series of subgroups of its Galois group, each one normal its successor with abelian quotient, or its Galois group is solvable. This proved to be a fertile approach, which later mathematicians adapted to many other fields of mathematics besides the theory of equations which Galois originally applied it to.[17]

Analysis

Galois also made some contributions to the theory of Abelian integrals and continued fractions.

See also

Notes

  1. ^ Stewart, J., et al. Algebra and Trigonometry. p. 334.
  2. ^ a b c Stewart, Ian (1973). Galois Theory. London: Chapman and Hall. pp. xvii-xxii. ISBN 0412108003.
  3. ^ Galois, Évariste (1828). "Démonstration d'un théorème sur les fractions continues périodiques". Annales de Mathématiques XIX: 294. http://www.numdam.org/item?id=AMPA_1828-1829__19__294_0.
  4. ^ a b c d Rothman, Tony. "Genius and Biographers: The Fictionalization of Evariste Galois". http://www.physics.princeton.edu/~trothman/galois.html. Retrieved 2009-02-04.
  5. ^ a b c Bell, Eric Temple (1986). Men of Mathematics. New York: Simon and Schuster. ISBN 0-671-62818-6.
  6. ^ a b c d Dupuy, Paul. "La vie d'Évariste Galois". Annales de l'École Normale 13: 197–266.
  7. ^ Galois, Évariste (1830). "Analyse d'un Mémoire sur la résolution algébrique des équations". Bulletin des Sciences mathématiques XIII: 271.
  8. ^ Galois, Évariste (1830). "Note sur la résolution des équations numériques". Bulletin des Sciences mathématiques XIII: 413.
  9. ^ a b Galois, Évariste (1830). "Sur la théorie des nombres". Bulletin des Sciences mathématiques XIII: 428.
  10. ^ a b Dumas (pére), Alexandre. "CCIV". Mes Mémoires. ISBN 1437155952. http://www.dumaspere.com/pages/bibliotheque/chapitre.php?lid=m3&cid=204. Retrieved 2010-04-13.
  11. ^ Taton, R. (1947). "Les relations d'Évariste Galois avec les mathématiciensde son temps". Revue d'histoire des sciences et de leurs applications 1: 114. doi:10.3406/rhs.1947.2607.
  12. ^ Bourgne, R.; J.-P. Azra (1962). Écrits et mémoires mathématiques d'Évariste Galois. Paris: Gauthier-Villars.
  13. ^ a b c Galois, Évariste (1846). "Lettre de Galois à M. Auguste Chevalier". Journal des mathématiques pures et appliquées XI: 408–415. http://visualiseur.bnf.fr/ark:/12148/cb343487840/date1846. Retrieved 2009-02-04.
  14. ^ Escofier, Jean-Pierre (2001). Galois Theory. Springer. pp. 224. ISBN 0387987657.
  15. ^ Rigatelli, Laura Toti (1996). Evariste Galois, 1811-1832 (Vita mathematica, 11). Birkhäuser. pp. 114. ISBN 3764354100.
  16. ^ Galois, Évariste (1846). "OEuvres mathématiques d'Évariste Galois.". Journal des mathématiques pures et appliquées XI: 381–444. http://visualiseur.bnf.fr/ark:/12148/cb343487840/date1846. Retrieved 2009-02-04.
  17. ^ a b Lie, Sophus (1895). "Influence de Galois sur le Développement des Mathématiques". Le centenaire de l'École Normale 1795-1985. Hachette.

References

External links

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