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| {{redirect|Fredrik Størmer|the engineer|Henrik Christian Fredrik Størmer|the jazz musician|Fredrik Carl Størmer}}
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| {{Infobox scientist
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| | birth_name = Fredrik Carl Mulertz Størmer
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| |name = Carl Størmer
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| |image =Portrett av Carl Størmer.jpg
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| |image_size =
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| |caption =
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| |birth_date = {{birth date|1874|9|3|df=y}}
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| |birth_place = [[Skien]]
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| |death_date = {{death date and age|1957|8|13|1874|9|3|df=y}}
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| |death_place = [[Oslo]]
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| |residence =
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| |citizenship =
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| |nationality = [[Norway|Norwegian]]
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| |ethnicity =
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| |field =[[mathematics]]
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| |work_institutions =
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| |alma_mater = [[University of Oslo]]
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| |doctoral_advisor =
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| |known_for = [[number theory]]<br/>[[Aurora (astronomy)|aurorae]]
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| |prizes = [[Fellow of the Royal Society]]<ref name="frs">{{cite doi|10.1098/rsbm.1958.0021|noedit}}</ref>
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| '''Fredrik Carl Mülertz Størmer''' (3 September 1874 – 13 August 1957) was a [[Norway|Norwegian]] [[mathematician]] and [[physicist]], known both for his work in [[number theory]] and for studying the movement of charged particles in the [[magnetosphere]] and the formation of [[Aurora (astronomy)|aurorae]].<ref name="brun">{{cite doi|10.1007/BF02559599}}</ref><ref name="mactutor">{{MacTutor|id=Stormer|title=Fredrik Carl Mülertz Størmer}}</ref>
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| ==Personal life and career==
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| Størmer was born on 3 September 1874 in [[Skien]], the only child of a pharmacist Georg Ludvig Størmer (1842–1930) and Elisabeth Amalie Johanne Henriette Mülertz (1844–1916).<ref name=nbl>{{cite encyclopedia|last=Grøn|first=Øyvind|authorlink=Øyvind Grøn|title=Carl Størmer|url=http://www.snl.no/.nbl_biografi/Carl_St%C3%B8rmer/utdypning|editor-last=Helle|editor-first=Knut|editor-link=Knut Helle|encyclopedia=[[Norsk biografisk leksikon]]|year=2004|publisher=Kunnskapsforlaget|location=Oslo|accessdate=25 December 2011|separator=,}}</ref> His uncle was the entrepreneur and inventor [[Henrik Christian Fredrik Størmer]].<ref name=nbl>{{Cite encyclopedia | first = S. | last = Haveraaen | year = 2004 | url = http://www.snl.no/.nbl_biografi/Fredrik_St%C3%B8rmer/utdypning | title = Fredrik Størmer | editor-last = Helle | editor-first = K. | editor-link = Knut Helle | encyclopedia = [[Norsk biografisk leksikon]] | publisher=[[Kunnskapsforlaget]] | location = Oslo | accessdate = 7 March 2011 | language = Norwegian}}</ref>
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| He studied mathematics at the [[University of Oslo|Royal Frederick University]] (now: University of Oslo) from 1892 to 1897, earning the rank of [[cand.real.|candidatus realium]] in 1898. He then studied with [[Charles Émile Picard|Picard]], [[Henri Poincaré|Poincaré]], [[Paul Painlevé|Painlevé]], [[Camille Jordan|Jordan]], [[Jean Gaston Darboux|Darboux]], and [[Edouard Goursat|Goursat]] at the [[University of Paris|Sorbonne]] in [[Paris]] from 1898 to 1900. On his return to Kristiania in 1900 as a [[research fellow]] in mathematics, he married Ada Clauson, with whom he eventually had five children. He visited the [[University of Göttingen]] in 1902, and returned to Kristiania in 1903, where he was appointed as a professor of mathematics, a position he held for 43 years. After he received a permanent position in Kristiania, Størmer published his subsequent writings under a shortened version of his name, Carl Størmer. In 1918, he was elected as the first president of the newly formed [[Norwegian Mathematical Society]]. He participated regularly in Scandinavian mathematical congresses, and was president of the 1936 [[International Congress of Mathematicians]] in Oslo (from 1924 the new name of Kristiania). Størmer was also affiliated with the Institute of Theoretical Astrophysics at the University of Oslo, which was founded in 1934. He died on 13 August 1957, at [[Blindern]].<ref name="frs"/><ref name="brun"/><ref name="mactutor"/><ref name="uio">{{cite web |url=http://www.fys.uio.no/plasma/historie/stormer/ |title=Fredrik Carl Mülertz Størmer |language=Norwegian |first=Alf |last=Egeland |publisher=Institute of Physics, [[University of Oslo]] |archivedate=26 May 2011 |archiveurl=http://web.archive.org/web/20080505005456/http://www.fys.uio.no/plasma/historie/stormer/ |accessdate=5 May 2008 |separator=,}}</ref>
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| Størmer was also an amateur [[street photography|street photographer]],<ref name="camera">{{cite book |title=80 millioner bilder: Norsk kulturhistorisk fotografi 1855–2005 |editor1-first=Jonas |editor1-last=Ekeberg |editor2-first=Harald Østgaard |editor2-last=Lund |year=2008 |publisher=Forlaget Press |location=Oslo |isbn=978-82-7547-315-6 |language=Norwegian |pages=142–45 |separator=,}}</ref> beginning in his student days, and near the age of 70 he put on an exhibition in Oslo of the photographs of celebrities that he had taken over the years.<ref name="frs"/> He was also a supervisory council member of the insurance company [[Forsikringsselskapet Norden]].<ref>{{cite news|title=Forsikringsaktieselskapet Nordens overskudd 377,000 kroner|date=2 May 1941|work=[[Aftenposten Aften]]|page=5}}</ref>
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| In February 1900 he married consul's daughter Ada Clauson (1877–1973). They had the son [[Leif Størmer]], who became a professor of [[historical geology]] at the [[University of Oslo]].<ref name=nbl/> His daughter Henny married landowner [[Carl Otto Løvenskiold (1898–1969)|Carl Otto Løvenskiold]].<ref>{{cite encyclopedia|year=1948|title=Løvenskiold, Carl Otto|encyclopedia=[[Hvem er hvem?]]|editor=Steenstrup, Bjørn|publisher=Aschehoug|location=Oslo|url=http://runeberg.org/hvemerhvem/1948/0363.html|page=344|language=Norwegian|accessdate=25 February 2012}}</ref> Carl Størmer is also the grandfather of the mathematician [[Erling Størmer]].<ref>{{cite encyclopedia|year=|last=Vatne|first=Jon Eivind|title=Erling Størmer|encyclopedia=[[Store norske leksikon]]|editor-last=Godal|editor-first=Anne Marit|publisher=Kunnskapsforlaget|location=Oslo|url=http://snl.no/Erling_St%C3%B8rmer|language=Norwegian|accessdate=27 May 2011|separator=,}}</ref><ref>{{cite encyclopedia|year=|title=Leif Størmer|encyclopedia=[[Store norske leksikon]]|editor-last=Godal|editor-first=Anne Marit|publisher=Kunnskapsforlaget|location=Oslo|url=http://snl.no/Leif_St%C3%B8rmer|language=Norwegian|accessdate=29 October 2009|separator=,}}</ref>
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| ==Mathematical research==
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| Størmer's first mathematical publication, published when he was a beginning student at the age of 18, concerned [[trigonometry|trigonometric]] [[series (mathematics)|series]] generalizing the [[Taylor expansion]] of the [[arcsine]] function, a problem he revisited again a few years later. Next, he systematically investigated [[Machin-like formula]] by which the number [[Pi|π]] may be represented as a rational combination of the so-called "[[Gregory number]]s" of the form tan<sup>−1</sup>(1/''n''). [[John Machin|Machin's]] original formula,
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| :<math>\textstyle\pi = 16\arctan \frac15 - 4\arctan\frac1{239},</math>
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| is of this type, and Størmer showed that there were three other ways of representing π as a rational combination of two Gregory numbers. He then investigated combinations of three Gregory numbers, and found 102 representations of π of this form, but was unable to determine whether there might be additional solutions of this type.<ref name="brun"/> These representations led to fast algorithms for computing [[numerical approximations of π]]; a four-term representation found by Størmer,
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| :<math>\textstyle\pi = 176 \arctan\frac{1}{57} + 28 \arctan\frac{1}{239} - 48 \arctan\frac{1}{682} + 96 \arctan\frac{1}{12943}</math>
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| was used in a record-setting calculation of π to 1,241,100,000,000 decimal digits in 2002 by [[Yasumasa Kanada]].<ref>Associated press, December 6, 2002; letter from Kanada, October 20, 2005, [http://www.super-computing.org/pi_current.html online at super-computing.org].</ref> Størmer is also noted for the [[Størmer number]]s, which arose from the decomposition of Gregory numbers in Størmer's work.
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| [[Størmer's theorem]], which he proved in 1897, shows that, for any finite set ''P'' of [[prime numbers]], there are only finitely many pairs of consecutive [[integers]] having only the numbers from ''P'' as their [[prime factor]]s. In addition, Størmer describes an [[algorithm]] for finding all such pairs. The [[superparticular ratio]]s generated by these consecutive pairs are of particular importance in music theory.<ref>{{cite journal |last1=Halsey |first1=G. D. |last2=Hewitt |first2=Edwin |author2-link=Edwin Hewitt |title=More on the superparticular ratios in music |journal=[[American Mathematical Monthly]] |volume=79 |year=1972 |pages=1096–1100 |mr=0313189 |doi=10.2307/2317424 |jstor=2317424 |issue=10 |publisher=Mathematical Association of America |separator=,}}</ref> Størmer proves this theorem by reducing the problem to a finite set of [[Pell's equation|Pell equations]], and the theorem itself can also be interpreted as describing the possible factorizations of solutions to Pell's equation. Chapman quotes [[Louis Mordell]] as saying "His result is very pretty, and there are many applications of it."<ref name="frs"/>
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| Additional subjects of Størmer's mathematical research included [[Lie group]]s, the [[gamma function]], and [[Diophantine approximation]] of [[algebraic number]]s and of the [[transcendental number]]s arising from [[elliptic function]]s. From 1905 Størmer was an editor of the journal ''[[Acta Mathematica]]'', and he was also an editor of the posthumously-published mathematical works of [[Niels Henrik Abel]] and [[Sophus Lie]].<ref name="frs"/><ref name="brun"/>
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| ==Astrophysical research==
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| [[File:Størmer and Birkeland Alta 1910.jpg|thumb|Carl Størmer with assistant Birkeland in 1910]]
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| From 1903, when Størmer first observed [[Kristian Birkeland]]'s experimental attempts to explain the [[Aurora (astronomy)|aurora borealis]], he was fascinated by aurorae and related phenomena. His first work on the subject attempted to model mathematically the paths taken by charged particles perturbed by the influence of a [[magnet]]ized [[sphere]], and Størmer eventually published over 48 papers on the motion of charged particles.<ref name="frs"/> By modeling the problem using [[differential equation]]s and [[polar coordinate]]s, Størmer was able to show that the [[Radius of curvature (mathematics)|radius of curvature]] of any particle's path is proportional to the square of its distance from the sphere's center. To solve the resulting differential equations numerically, he used [[Verlet integration]], which is therefore also known as ''Störmer's method''.<ref name="brun"/> [[Ernst Brüche]] and [[Willard Harrison Bennett]] verified experimentally Størmer's predicted particle motions; Bennett called his experimental apparatus "Störmertron" in honor of Størmer.<ref name="frs"/> Størmer's calculations showed that small variations in the trajectories of particles approaching the earth would be magnified by the effects of the Earth's magnetic field, explaining the convoluted shapes of aurorae.<ref>{{cite journal |first=P. G. |last=Nutting |authorlink=Perley G. Nutting |title=Störmer's work on the physics of the aurora |journal=[[Journal of Geophysical Research|Terrestrial Magnetism and Atmospheric Electricity]] |year=1908 |doi=10.1029/TE013i001p00023 |volume=13 |pages=23 |bibcode=1908TeMAE..13...23N |separator=,}}</ref> Størmer also considered the possibility that particles might be trapped within the magnetic field, and worked out the orbits of these trapped particles, a prediction that was borne out after his death by the 1958 discovery of the [[Van Allen radiation belt]].<ref>{{cite journal |title=Energetic particles in the inner Van Allen belt |journal=Space Science Reviews |first=Wilmot N. |last=Hess |authorlink=Wilmot N. Hess |volume=1 |issue=2 |year=1962 |pages=278–312 |doi=10.1007/BF00240580 |bibcode=1962SSRv....1..278H |separator=,}}</ref>
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| As well as modeling these phenomena mathematically, Størmer took many [[photograph]]s of aurorae, from 20 different observatories across Norway. He measured their heights and latitudes by [[triangulation]] from multiple observatories, and discovered that the aurora are typically as high as 100 kilometers above ground. He classified them by their shapes, and discovered in 1926 the "solar-illuminated aurora", a phenomenon that can occur at twilight when the upper parts of an aurora are lit by the sun; these aurorae can be as high as 1000 km above ground.<ref>{{cite web |url=http://www.northern-lights.no/english/science/stormer.shtml |archiveurl=http://web.archive.org/web/20081201233614/http://www.northern-lights.no/english/science/stormer.shtml |title=Størmer |year=2003 |publisher=Northern Lights Center |archivedate=26 May 2011 |accessdate=1 December 2008 |separator=,}}</ref><ref>{{cite web |url=http://image.gsfc.nasa.gov/poetry/tour/bioStorm.html |title=Carl Stormer |publisher=[[NASA]] IMAGE Education Center |accessdate=26 May 2011 |archiveurl=http://www.webcitation.org/5yyg2zTNQ |archivedate=26 May 2011 |separator=,}}</ref> His work, including laboratory 'aurora reproductions', had gathered international popular interest by 1928.<ref>{{Cite journal |date=November 1928 |title=Auroras shot from the Sun |journal=Popular Science |publisher=Bonnier Corporation |volume=113 |issue=5 |pages=58 |issn=0161-7370 |separator=,}}</ref>
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| Størmer's book, ''From the depths of space to the heart of the atom'', describing his work in this area, was translated into five different languages from the original Norwegian.<ref name="brun"/> A second book, ''The Polar Aurora'' (Oxford Press, 1955), contains both his experimental work on aurorae and his mathematical attempts to model them. In his review of this book,<ref>{{cite journal |title=The Polar Aurora by Carl Størmer |first=J. F. |last=Heard |journal=Journal of the Royal Astronomical Society of Canada |volume=51 |pages=117–118 |separator=,}}</ref> J. F. Heard calls Størmer "the acknowledged authority" on aurorae. Heard writes, "''The Polar Aurora'' will undoubtedly remain for many years a standard reference book; it belongs on the desk of anyone whose work or interest is involved with aurorae."
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| Other astrophysical phenomena investigated by Størmer include pulsations of the [[earth's magnetic field]], [[Long delayed echo|echoing]] in [[radio]] transmissions, [[nacreous cloud]]s and luminous night clouds, [[zodiacal light]], [[meteor]] trails, the [[solar corona]] and solar vortices, and [[cosmic ray]]s.<ref name="frs"/>
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| ==Awards and honors==
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| Størmer was a [[Royal Society|Foreign Member of the Royal Society (ForMemRS]]<ref name="frs"/> and a corresponding member of the [[French Academy of Sciences]].<ref name="frs"/> He was also a member of the [[Norwegian Academy of Science and Letters]] from 1900.<ref name=nbl/> He was given [[honorary degree]]s by [[Oxford University]] (in 1947), the [[University of Copenhagen]] (1951), and the [[University of Paris|Sorbonne]] (1953), and in 1922 the French Academy awarded him their [[Janssen Medal (French Academy of Sciences)|Janssen Medal]].<ref name="frs"/><ref name="mactutor">{{cite web |url=http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Stormer.html |title=Fredrik Carl Mulertz Stormer |publisher=MacTutor history of mathematics archive |accessdate=26 May 2011 |archiveurl=http://www.webcitation.org/5yygItzO6 |archivedate=26 May 2011 |separator=,}}</ref><ref name=nbl/> Three times Størmer was a plenary speaker in the International Congress of
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| Mathematicians.<ref>{{cite web|title=ICM Plenary and Invited speakers|url= http://www.mathunion.org/db/ICM/Speakers/SortedByLastname.php}}</ref> In 1971, the crater [[Störmer (crater)|Störmer]] on the far side of the Moon was named after him.<ref>{{cite journal |year=1971 |title=Report on Lunar Nomenclature by The Working Group of Commission 17 of the IAU |journal=Space Science Reviews |volume=12 |issue=2 |pages=136–186 |doi=10.1007/BF00171763 |last1=Menzel |first1=D. H. |author1-link=Donald Howard Menzel |last2=Minnaert |first2=M. |author2-link=Marcel Minnaert |last3=Levin |first3=B. |last4=Dollfus |first4=A. |author4-link=Audouin Dollfus |last5=Bell |first5=B. |bibcode=1971SSRv...12..136M |separator=,}}</ref>
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| In 1902, Størmer was decorated with [[King Oscar II]]'s Medal of Merit in gold. He was also decorated as a Knight, First Order of the [[Order of St. Olav]] in 1939. He was upgraded to [[Grand Cross]] of the Order of St. Olav in 1954.<ref name=nbl/>
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| ==References==
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| {{Reflist|2}}
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| {{Authority control|VIAF=17366391}}
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| {{Persondata <!-- Metadata: see [[Wikipedia:Persondata]]. -->
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| | NAME = Stormer, Carl
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| | ALTERNATIVE NAMES =
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| | SHORT DESCRIPTION = norwegian astronomer
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| | DATE OF BIRTH = September 3, 1874
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| | PLACE OF BIRTH = [[Skien]]
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| | DATE OF DEATH = August 13, 1957
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| | PLACE OF DEATH = [[Oslo]]
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| }}
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| {{DEFAULTSORT:Stormer, Carl}}
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| [[Category:1874 births]]
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| [[Category:1957 deaths]]
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| [[Category:People from Skien]]
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| [[Category:University of Oslo alumni]]
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| [[Category:University of Paris alumni]]
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| [[Category:University of Oslo faculty]]
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| [[Category:Norwegian astronomers]]
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| [[Category:Norwegian mathematicians]]
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| [[Category:Norwegian physicists]]
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| [[Category:Number theorists]]
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| [[Category:20th-century astronomers]]
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| [[Category:Members of the Norwegian Academy of Science and Letters]]
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| [[Category:Foreign Members of the Royal Society]]
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| [[Category:Members of the French Academy of Sciences]]
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| [[Category:Knights Grand Cross of the Order of St. Olav]]
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