Editing Speed of light (section)

== Measurement ==
<!--- The article Galileo Galileo links to this section. Please do not change the title of the section without amending the articles which link to it. --->
There are different ways to determine the value of ''c''. One way is to measure the actual speed at which light waves propagate, which can be done in various astronomical and Earth-based setups. It is also possible to determine ''c'' from other physical laws where it appears, for example, by determining the values of the electromagnetic constants [[relative permittivity|''ε''<sub>0</sub>]] and [[permeability (electromagnetism)|''μ''<sub>0</sub>]] and using their relation to ''c''. Historically, the most accurate results have been obtained by separately determining the frequency and wavelength of a light beam, with their product equalling ''c''. This is described in more detail in the [[#Interferometry|"Interferometry" section]] below.

In 1983 the metre was defined as "the length of the path travelled by light in vacuum during a time interval of {{frac|1|{{val|299792458}}}} of a second",<ref name=Resolution_1/> fixing the value of the speed of light at {{val|299792458|u=m/s}} by definition, as [[#Increased accuracy of c and redefinition of the metre and second|described below]]. Consequently, accurate measurements of the speed of light yield an accurate realization of the metre rather than an accurate value of ''c''.

=== Astronomical measurements ===
[[File:Io eclipse speed of light measurement.svg|thumb|upright=1.8|Measurement of the speed of light from the time it takes Io to orbit Jupiter, using eclipses of Io by Jupiter's shadow to precisely measure its orbit.]]
[[Outer space]] is a convenient setting for measuring the speed of light because of its large scale and nearly perfect [[vacuum]]. Typically, one measures the time needed for light to traverse some reference distance in the [[Solar System]], such as the [[radius]] of the Earth's orbit. Historically, such measurements could be made fairly accurately, compared to how accurately the length of the reference distance is known in Earth-based units.

[[Ole Rømer]] used an astronomical measurement to make [[Rømer's determination of the speed of light|the first quantitative estimate of the speed of light]] in the year 1676.<ref name=cohen>
{{Cite journal
 |last=Cohen |first=I. B. |author-link=I. Bernard Cohen
 |year=1940
 |title=Roemer and the first determination of the velocity of light (1676)
 |journal=[[Isis (journal)|Isis]]
 |volume=31 |issue=2 |pages=327–379
 |doi=10.1086/347594
 |ref=cohen-1940
 |hdl=2027/uc1.b4375710
 |s2cid=145428377
 |url=https://babel.hathitrust.org/cgi/imgsrv/download/pdf?id=uc1.b4375710;orient=0;size=100;seq=5;attachment=0
 |hdl-access=free
}}</ref><ref name=roemer>
{{Cite journal
 |year=1676
 |title=Demonstration tovchant le mouvement de la lumiere trouvé par M. Rŏmer de l'Académie Royale des Sciences
 |trans-title=Demonstration to the movement of light found by Mr. Römer of the Royal Academy of Sciences
 |language=fr
 |url=http://www-obs.univ-lyon1.fr/labo/fc/ama09/pages_jdsc/pages/jdsc_1676_lumiere.pdf
 |journal=[[Journal des sçavans]]
 |pages=233–236
 |ref=roemer-1676
}}<br />Translated in 
{{Cite journal
 |doi=10.1098/rstl.1677.0024
 |year=1677
 |title=A demonstration concerning the motion of light, communicated from Paris, in the Journal des Sçavans, and here made English
 |journal=[[Philosophical Transactions of the Royal Society]]
 |volume=12 |issue=136 |pages=893–895
 |ref=roemer-1676-EnglishTrans
 |bibcode=1677RSPT...12..893.|doi-access=free
}}<br />Reproduced in 
{{Cite book
 |editor1-last=Hutton |editor1-first=C.
 |editor2-last=Shaw |editor2-first=G.
 |editor3-last=Pearson |editor3-first=R.
 |year=1809
 |title=The Philosophical Transactions of the Royal Society of London, from Their Commencement in 1665, in the Year 1800: Abridged
 |chapter=On the Motion of Light by M. Romer
 |chapter-url=https://archive.org/stream/philosophicaltra02royarich#page/397/mode/1up
 |location=London |publisher=C. & R. Baldwin
 |volume=II. From 1673 to 1682 |pages=397–398
}}<br />
The account published in {{lang|fr|Journal des sçavans}} was based on a report that Rømer read to the [[French Academy of Sciences]] in November 1676 [[#cohen-1940|(Cohen, 1940, p.&nbsp;346)]].</ref> When measured from Earth, the periods of moons orbiting a distant planet are shorter when the Earth is approaching the planet than when the Earth is receding from it. The difference is small, but the cumulative time becomes significant when measured over months. The distance travelled by light from the planet (or its moon) to Earth is shorter when the Earth is at the point in its orbit that is closest to its planet than when the Earth is at the farthest point in its orbit, the difference in distance being the [[diameter]] of the Earth's orbit around the Sun. The observed change in the moon's orbital period is caused by the difference in the time it takes light to traverse the shorter or longer distance. Rømer observed this effect for [[Jupiter]]'s innermost major moon Io and deduced that light takes 22 minutes to cross the diameter of the Earth's orbit.<ref name="cohen" />

[[File:SoL Aberration.svg|thumb|upright|Aberration of light: light from a distant source appears to be from a different location for a moving telescope due to the finite speed of light.|alt=A star emits a light ray that hits the objective of a telescope. While the light travels down the telescope to its eyepiece, the telescope moves to the right. For the light to stay inside the telescope, the telescope must be tilted to the right, causing the distant source to appear at a different location to the right.]]
Another method is to use the [[aberration of light]], discovered and explained by [[James Bradley]] in the 18th century.<ref name="Bradley1729">
{{Cite journal
 |last=Bradley |first=J.
 |year=1729
 |title=Account of a new discovered Motion of the Fix'd Stars
 |url=http://gallica.bnf.fr/ark:/12148/bpt6k55840n.image.f375.langEN
 |journal=[[Philosophical Transactions]]
 |volume=35 |pages=637–660
}}</ref> This effect results from the [[vector addition]] of the velocity of light arriving from a distant source (such as a star) and the velocity of its observer (see diagram on the right). A moving observer thus sees the light coming from a slightly different direction and consequently sees the source at a position shifted from its original position. Since the direction of the Earth's velocity changes continuously as the Earth orbits the Sun, this effect causes the apparent position of stars to move around. From the angular difference in the position of stars (maximally 20.5 [[arcsecond]]s)<ref>
{{Cite book
 |last=Duffett-Smith
 |first=P.
 |year=1988
 |title=Practical Astronomy with your Calculator
 |url=https://archive.org/details/practicalastrono0000duff
 |url-access=registration
 |page=[https://archive.org/details/practicalastrono0000duff/page/62 62]
 |publisher=Cambridge University Press
 |isbn=978-0-521-35699-2
}} [https://archive.org/details/practicalastrono0000duff/page/62 Extract of page 62].</ref> it is possible to express the speed of light in terms of the Earth's velocity around the Sun, which with the known length of a year can be converted to the time needed to travel from the Sun to the Earth. In 1729, Bradley used this method to derive that light travelled {{val|10,210}} times faster than the Earth in its orbit (the modern figure is {{val|10,066}} times faster) or, equivalently, that it would take light 8&nbsp;minutes 12&nbsp;seconds to travel from the Sun to the Earth.<ref name="Bradley1729"/>

==== Astronomical unit ====
{{main|Astronomical unit}}
Historically the speed of light was used together with timing measurements to determine a value for the  astronomical unit (AU).<ref>{{Cite journal |last=Standish |first=E. M. |date=June 2004 |title=The Astronomical Unit now |url=https://www.cambridge.org/core/product/identifier/S1743921305001365/type/journal_article |journal=Proceedings of the International Astronomical Union |language=en |volume=2004 |issue=IAUC196 |pages=163–179 |doi=10.1017/S1743921305001365 |issn=1743-9213}}</ref> It was redefined in 2012 as exactly {{val|149597870700|u=m}}.<ref>{{Cite journal|journal=The International System of Units|title=Supplement 2014: Updates to the 8th edition (2006) of the SI Brochure|url=http://www.bipm.org/utils/common/pdf/si_supplement_2014.pdf|year=2014|publisher= International Bureau of Weights and Measures|page=14}}</ref><ref name=AU_redef>{{Cite web|title=Resolution B2 on the re-definition of the astronomical unit of length|url=https://www.iau.org/static/resolutions/IAU2012_English.pdf|year=2012|publisher=International Astronomical Union}}</ref> This redefinition is analogous to that of the metre and likewise has the effect of fixing the speed of light to an exact value in astronomical units per second (via the exact speed of light in metres per second).<ref>{{Cite journal|last=Brumfiel|first=Geoff|date=14 September 2012|title=The astronomical unit gets fixed|url=https://www.nature.com/articles/nature.2012.11416|journal=[[Nature (journal)|Nature]]|language=en|doi=10.1038/nature.2012.11416|s2cid=123424704|issn=1476-4687|url-access=subscription}}</ref>

=== Time of flight techniques ===
[[File:Michelson speed of light measurement 1930.jpg|thumb|center|One of the last and most accurate time of flight measurements, Michelson, Pease and Pearson's 1930–1935 experiment used a rotating mirror and a one-mile (1.6&nbsp;km) long vacuum chamber which the light beam traversed 10 times. It achieved accuracy of ±11&nbsp;km/s.|600x600px]]
[[File:Fizeau-int.svg|thumb|Diagram of the [[Fizeau's measurement of the speed of light in air|Fizeau apparatus]]:{{image key|list type=ordered
|Light source
|Beam-splitting semi-transparent mirror
|Toothed wheel-breaker of the light beam
|Remote mirror
|Telescopic tube}}|alt=A light ray passes horizontally through a half-mirror and a rotating cog wheel, is reflected back by a mirror, passes through the cog wheel, and is reflected by the half-mirror into a monocular.]]

A method of measuring the speed of light is to measure the time needed for light to travel to a mirror at a known distance and back. This is the working principle behind experiments by [[Hippolyte Fizeau]] and [[Léon Foucault]].

The [[Fizeau's measurement of the speed of light in air|setup as used by Fizeau]] consists of a beam of light directed at a mirror {{convert|8|km|mi|0}} away. On the way from the source to the mirror, the beam passes through a rotating cogwheel. At a certain rate of rotation, the beam passes through one gap on the way out and another on the way back, but at slightly higher or lower rates, the beam strikes a tooth and does not pass through the wheel. Knowing the distance between the wheel and the mirror, the number of teeth on the wheel, and the rate of rotation, the speed of light can be calculated.<ref name=How>
{{Cite web
 |last=Gibbs
 |first=P.
 |year=1997
 |title=How is the speed of light measured?
 |url=http://math.ucr.edu/home/baez/physics/Relativity/SpeedOfLight/measure_c.html
 |work=Usenet Physics FAQ
 |publisher=University of California, Riverside
 |access-date=13 January 2010
 |url-status=dead
 |archive-url=https://web.archive.org/web/20150821181850/http://math.ucr.edu/home/baez/physics/Relativity/SpeedOfLight/measure_c.html
 |archive-date=21 August 2015
}}</ref>

The [[Foucault's measurements of the speed of light|method of Foucault]] replaces the cogwheel with a rotating mirror. Because the mirror keeps rotating while the light travels to the distant mirror and back, the light is reflected from the rotating mirror at a different angle on its way out than it is on its way back. From this difference in angle, the known speed of rotation and the distance to the distant mirror the speed of light may be calculated.<ref>
{{Cite web
 |last=Fowler |first=M.
 |title=The Speed of Light
 |url=http://galileoandeinstein.physics.virginia.edu/lectures/spedlite.html
 |publisher=University of Virginia
 |access-date=21 April 2010
}}</ref> Foucault used this apparatus to measure the speed of light in air versus water, based on a suggestion by [[François Arago]].<ref name="Hughes2012">{{Cite book|last1=Hughes|first1=Stephan|url=https://books.google.com/books?id=iZk5OOf7fVYC|title=Catchers of the Light: The Forgotten Lives of the Men and Women Who First Photographed the Heavens|date=2012|publisher=ArtDeCiel Publishing|isbn=978-1-62050-961-6|pages=210}}</ref>

Today, using [[oscilloscopes]] with time resolutions of less than one nanosecond, the speed of light can be directly measured by timing the delay of a light pulse from a laser or an LED reflected from a mirror. This method is less precise (with errors of the order of 1%) than other modern techniques, but it is sometimes used as a laboratory experiment in college physics classes.<ref>See, for example:
* {{Cite journal
 |last1=Cooke |first1=J.
 |last2=Martin |first2=M.
 |last3=McCartney |first3=H.
 |last4=Wilf |first4=B.
 |year=1968
 |title=Direct determination of the speed of light as a general physics laboratory experiment
 |journal=[[American Journal of Physics]]
 |volume=36 |issue=9 |page=847
 |doi=10.1119/1.1975166
 |bibcode = 1968AmJPh..36..847C
}}
* {{Cite journal
 |last1=Aoki |first1=K. |last2=Mitsui |first2=T.
 |year=2008
 |title=A small tabletop experiment for a direct measurement of the speed of light
 |journal=[[American Journal of Physics]]
 |volume=76 |issue=9 |pages=812–815
 |doi=10.1119/1.2919743
 |arxiv=0705.3996
 |bibcode = 2008AmJPh..76..812A |s2cid=117454437
}}
* {{Cite journal
 |last1=James |first1=M. B. |last2=Ormond |first2=R. B. |last3=Stasch |first3=A. J.
 |year=1999
 |title=Speed of light measurement for the myriad
 |journal=[[American Journal of Physics]]
 |volume=67 |issue=8 |pages=681–714
 |doi=10.1119/1.19352
 |bibcode = 1999AmJPh..67..681J
}}</ref>

=== Electromagnetic constants ===
An option for deriving ''c'' that does not directly depend on a measurement of the propagation of electromagnetic waves is to use the relation between ''c'' and the [[vacuum permittivity]] ''ε''<sub>0</sub> and [[vacuum permeability]] ''μ''<sub>0</sub> established by Maxwell's theory: ''c''<sup>2</sup>&nbsp;=&nbsp;1/(''ε''<sub>0</sub>''μ''<sub>0</sub>). The vacuum permittivity may be determined by measuring the [[capacitance]] and dimensions of a [[capacitor]], whereas the value of the vacuum permeability was historically fixed at exactly {{val|4|end=π|e=-7|u=H.m-1}} through the definition of the [[ampere (unit)|ampere]]. [[Edward Bennett Rosa|Rosa]] and [[Noah Ernest Dorsey|Dorsey]] used this method in 1907 to find a value of {{val|299710|22|u=km/s}}. Their method depended upon having a standard unit of electrical resistance, the "international [[ohm]]", and so its accuracy was limited by how this standard was defined.<ref name="Essen1948"/><ref name="RosaDorsey">{{Cite journal |last1=Rosa |first1=E. B. |author-link=Edward Bennett Rosa |last2=Dorsey |first2=N. E. |author-link2=Noah Ernest Dorsey |year=1907 |title=A new determination of the ratio of the electromagnetic to the electrostatic unit of electricity |journal=Bulletin of the Bureau of Standards |volume=3 |issue=6 |page=433 |doi=10.6028/bulletin.070 |doi-access=free}}</ref>

=== Cavity resonance ===
[[File:Waves in Box.svg|thumb|right|Electromagnetic [[standing waves]] in a cavity|alt=A box with three waves in it; there are one and a half wavelength of the top wave, one of the middle one, and a half of the bottom one.]]

Another way to measure the speed of light is to independently measure the frequency ''f'' and wavelength ''λ'' of an electromagnetic wave in vacuum. The value of ''c'' can then be found by using the relation ''c''&nbsp;=&nbsp;''fλ''. One option is to measure the resonance frequency of a [[cavity resonator]]. If the dimensions of the resonance cavity are also known, these can be used to determine the wavelength of the wave. In 1946, [[Louis Essen]] and A.C. Gordon-Smith established the frequency for a variety of [[normal mode]]s of microwaves of a [[microwave cavity]] of precisely known dimensions. The dimensions were established to an accuracy of about ±0.8&nbsp;μm using gauges calibrated by interferometry.<ref name="Essen1948"/> As the wavelength of the modes was known from the geometry of the cavity and from [[electromagnetic theory]], knowledge of the associated frequencies enabled a calculation of the speed of light.<ref name="Essen1948">
{{Cite journal
 |last1=Essen |first1=L.
 |last2=Gordon-Smith |first2=A. C.
 |year=1948
 |title=The Velocity of Propagation of Electromagnetic Waves Derived from the Resonant Frequencies of a Cylindrical Cavity Resonator
 |journal=[[Proceedings of the Royal Society of London A]]
 |volume=194 |issue=1038 |pages=348–361
 |doi=10.1098/rspa.1948.0085
 |bibcode=1948RSPSA.194..348E
 |jstor=98293
 |doi-access=free
}}</ref><ref>
{{Cite journal
 |last=Essen |first=L.
 |year=1947
 |title=Velocity of Electromagnetic Waves
 |journal=Nature
 |volume=159 |issue=4044 |pages=611–612
 |doi=10.1038/159611a0
 |bibcode=1947Natur.159..611E
 |s2cid=4101717
}}</ref>

The Essen–Gordon-Smith result, {{val|299792|9|u=km/s}}, was substantially more precise than those found by optical techniques.<ref name="Essen1948" /> By 1950, repeated measurements by Essen established a result of {{val|299792.5|3.0|u=km/s}}.<ref name="Essen1950">
{{Cite journal
 |last=Essen |first=L.
 |year=1950
 |title=The Velocity of Propagation of Electromagnetic Waves Derived from the Resonant Frequencies of a Cylindrical Cavity Resonator
 |journal=[[Proceedings of the Royal Society of London A]]
 |volume=204 |issue=1077 |pages=260–277
 |doi=10.1098/rspa.1950.0172
 |bibcode=1950RSPSA.204..260E
 |jstor=98433
 |s2cid=121261770
}}</ref>

A household demonstration of this technique is possible, using a [[microwave oven]] and food such as marshmallows or margarine: if the turntable is removed so that the food does not move, it will cook the fastest at the [[antinode]]s (the points at which the wave amplitude is the greatest), where it will begin to melt. The distance between two such spots is half the wavelength of the microwaves; by measuring this distance and multiplying the wavelength by the microwave frequency (usually displayed on the back of the oven, typically 2450&nbsp;MHz), the value of ''c'' can be calculated, "often with less than 5% error".<ref>
{{Cite journal
 |last = Stauffer | first = R. H.
 |date=April 1997
 |title = Finding the Speed of Light with Marshmallows
 |journal = [[The Physics Teacher]]
 |volume = 35
 |issue = 4 
 |page = 231
 |url = https://www.physics.umd.edu/icpe/newsletters/n34/marshmal.htm
 |access-date = 15 February 2010
 |bibcode = 1997PhTea..35..231S |doi = 10.1119/1.2344657
|url-access = subscription
 }}</ref><ref>
{{Cite web
 |url =http://www.bbc.co.uk/norfolk/features/ba_festival/bafestival_speedoflight_experiment_feature.shtml
 |title = BBC Look East at the speed of light
 |work = BBC Norfolk website
 |access-date = 15 February 2010
}}</ref>

=== Interferometry ===
[[File:Interferometer sol.svg|thumb|An interferometric determination of length. Left: [[constructive interference]]; Right: [[destructive interference]].|alt=Schematic of the working of a Michelson interferometer.]]
[[Interferometry]] is another method to find the wavelength of electromagnetic radiation for determining the speed of light.<ref name=Vaughan>
{{Cite book
 |last=Vaughan |first=J. M.
 |year=1989
 |title=The Fabry-Perot interferometer
 |url=https://books.google.com/books?id=mMLuISueDKYC
 |pages=47, 384–391
 |publisher=CRC Press
 |isbn=978-0-85274-138-2
}}</ref> A [[Coherence (physics)|coherent]] beam of light (e.g. from a [[laser]]), with a known frequency (''f''), is split to follow two paths and then recombined. By adjusting the path length while observing the [[interference (wave propagation)|interference pattern]] and carefully measuring the change in path length, the wavelength of the light (''λ'') can be determined. The speed of light is then calculated using the equation&nbsp;''c''&nbsp;=&nbsp;''λf''.

Before the advent of laser technology, coherent [[radiowave|radio]] sources were used for interferometry measurements of the speed of light.<ref name=Froome1858>
{{Cite journal
 |doi=10.1098/rspa.1958.0172
 |title=A New Determination of the Free-Space Velocity of Electromagnetic Waves
 |first=K. D.
 |last=Froome
 |journal=Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences
 |volume=247
 |year=1958
 |pages=109–122
 |issue=1248
 |bibcode = 1958RSPSA.247..109F
 |jstor=100591 |s2cid=121444888
}}</ref> Interferometric determination of wavelength becomes less precise with wavelength and the experiments were thus limited in precision by the long wavelength (~{{cvt|4|mm|in}}) of the radiowaves. The precision can be improved by using light with a shorter wavelength, but then it becomes difficult to directly measure the frequency of the light.<ref name="NIST_pub"/>

One way around this problem is to start with a low frequency signal of which the frequency can be precisely measured, and from this signal progressively synthesize higher frequency signals whose frequency can then be linked to the original signal. A laser can then be locked to the frequency, and its wavelength can be determined using interferometry.<ref name="NIST_pub">
{{Cite book
 |title        = A Century of Excellence in Measurements, Standards, and Technology
 |editor-last  = Lide
 |editor-first = D. R.
 |contribution = Speed of Light from Direct Frequency and Wavelength Measurements
 |last         = Sullivan
 |first        = D. B.
 |year         = 2001
 |pages        = 191–193
 |publisher    = CRC Press
 |isbn         = 978-0-8493-1247-2
 |url          = http://nvl.nist.gov/pub/nistpubs/sp958-lide/191-193.pdf
 |url-status   = dead
 |archive-url  = https://web.archive.org/web/20090813061018/http://nvl.nist.gov/pub/nistpubs/sp958-lide/191-193.pdf
 |archive-date = 13 August 2009
}}</ref> This technique was due to a group at the National Bureau of Standards (which later became the [[National Institute of Standards and Technology]]). They used it in 1972 to measure the speed of light in vacuum with a [[Measurement uncertainty|fractional uncertainty]] of {{val|3.5|e=-9}}.<ref name="NIST_pub"/><ref name="NIST heterodyne">
{{Cite journal
 |last1=Evenson |first1=K. M. |year=1972
 |title=Speed of Light from Direct Frequency and Wavelength Measurements of the Methane-Stabilized Laser
 |journal=Physical Review Letters
 |volume=29
 |issue=19 |pages=1346–1349
 |doi=10.1103/PhysRevLett.29.1346
 |bibcode=1972PhRvL..29.1346E
 |s2cid=120300510 |display-authors=etal
}}</ref>