Rømer's determination of the speed of light
Rømer's determination of the speed of light was the demonstration in 1676 that light has a finite speed and so does not travel instantaneously. The discovery is usually attributed to Danish astronomer Ole Rømer (1644–1710),[note 1] who was working at the Royal Observatory in Paris at the time.
By timing the eclipses of the Jupiter moon Io, Rømer estimated that light would take about 22 minutes to travel a distance equal to the diameter of Earth's orbit around the Sun. This would give light a velocity of about 220,000 kilometres per second, about 26% lower than the true value of 299,792 km/s.
Rømer's theory was controversial at the time he announced it, and he never convinced the director of the Paris Observatory, Giovanni Domenico Cassini, to fully accept it. However, it quickly gained support among other natural philosophers of the period, such as Christiaan Huygens and Isaac Newton. It was finally confirmed nearly two decades after Rømer's death, with the explanation in 1729 of stellar aberration by the English astronomer James Bradley.
The determination of east-west positioning (longitude) was a significant practical problem in cartography and navigation before the 1700s. In 1598 Philip III of Spain had offered a prize for a method to determine the longitude of a ship out of sight of land. Galileo proposed a method of establishing the time of day, and thus longitude, based on the times of the eclipses of the moons of Jupiter, in essence using the Jovian system as a cosmic clock; this method was not significantly improved until accurate mechanical clocks were developed in the eighteenth century. Galileo proposed this method to the Spanish crown (1616–17) but it proved to be impractical, not least because of the difficulty of observing the eclipses from a ship. However, with refinements the method could be made to work on land.
The Italian astronomer Giovanni Domenico Cassini had pioneered the use of the eclipses of the Galilean moons for longitude measurements, and published tables predicting when eclipses would be visible from a given location. He was invited to France by Louis XIV to set up the Royal Observatory, which opened in 1671 with Cassini as director, a post he would hold for the rest of his life.
One of Cassini's first projects at his new post in Paris was to send Frenchman Jean Picard to the site of Tycho Brahe's old observatory at Uraniborg, on the island of Hven near Copenhagen. Picard was to observe and time the eclipses of Jupiter's moons from Uraniborg while Cassini recorded the times they were seen in Paris. If Picard recorded the end of an eclipse at 9 hours 43 minutes 54 seconds after midday in Uraniborg, while Cassini recorded the end of the same eclipse at 9 hours 1 minute 44 seconds after midday in Paris – a difference of 42 minutes 10 seconds – the difference in longitude could be calculated to be 10° 32' 30".[note 2] Picard was helped in his observations by a young Dane who had recently completed his studies at the University of Copenhagen – Ole Rømer – and he must have been impressed by his assistant's skills, as he arranged for the young man to come to Paris to work at the Royal Observatory there.
Eclipses of Io
Io is the innermost of the four moons of Jupiter discovered by Galileo in January 1610. Rømer and Cassini refer to it as the "first satellite of Jupiter". It orbits Jupiter once every 42½ hours, and the plane of its orbit is very close to the plane of Jupiter's orbit around the sun. This means that it passes some of each orbit in the shadow of Jupiter – an eclipse.
Viewed from the Earth, an eclipse of Io is seen in one of two ways.
- Io suddenly disappears, as it moves into the shadow of Jupiter. This is termed an immersion.
- Io suddenly reappears, as it moves out of the shadow of Jupiter. This is called an emergence.
From the Earth, it is not possible to view both the immersion and the emergence for the same eclipse of Io, because one or the other will be hidden (occulted) by Jupiter itself. At the point of opposition (point H in the diagram below), both the immersion and the emergence would be hidden by Jupiter.
For about four months after the opposition of Jupiter (from L to K in the diagram below), it is possible to view emergences of Io from its eclipses, while for about four months before the opposition (from F to G), it is possible to view immersions of Io into Jupiter's shadow. For about five or six months of the year, around the point of conjunction, it is impossible to observe the eclipses of Io at all because Jupiter is too close (in the sky) to the sun. Even during the periods before and after opposition, not all of the eclipses of Io can be observed from a given location on the Earth's surface: some eclipses will occur during the daytime for a given location, while other eclipses will occur while Jupiter is below the horizon (hidden by the Earth itself).
The key phenomenon that Rømer observed was that the time elapsed between eclipses was not constant. Rather, it varied slightly at different times of year. Since he was fairly confident that the orbital period of Io was not actually changing, he deduced that this was an observational effect. The orbital paths of Earth and Jupiter being available to him, he noticed that periods in which Earth and Jupiter were moving away from each other always corresponded to a longer interval between eclipses. Conversely, the times when Earth and Jupiter were moving closer together were always accompanied by a decrease in the eclipse interval. This, Rømer reasoned, could be satisfactorily explained if light possessed a finite speed, which he went on to calculate.
Most of Rømer's papers were destroyed in the Copenhagen Fire of 1728, but one manuscript that survived contains a listing of about sixty observations of eclipses of Io from 1668 to 1678. In particular, it details two series of observations on either side of the oppositions of 2 March 1672 and 2 April 1673. Rømer comments in a letter to Christiaan Huygens dated 30 September 1677 that these observations from 1671–73 form the basis for his calculations.
The surviving manuscript was written some time after January 1678, the date of the last recorded astronomical observation (an emergence of Io on 6 January), and so was later than Rømer's letter to Huygens. Rømer appears to have been collecting data on eclipses of the Galilean moons in the form of an aide-mémoire, possibly as he was preparing to return to Denmark in 1681. The document also records the observations around the opposition of 8 July 1676 that formed the basis for the announcement of Rømer's results.
On 22 August 1676,[note 3] Cassini made an announcement to the Royal Academy of Sciences in Paris that he would be changing the basis of calculation for his tables of eclipses of Io. He may also have stated the reason:[note 4]
This second inequality appears to be due to light taking some time to reach us from the satellite; light seems to take about ten to eleven minutes [to cross] a distance equal to the half-diameter of the terrestrial orbit.
Most importantly, Rømer announced the prediction that the emergence of Io on 16 November 1676 would be observed about ten minutes later than would have been calculated by the previous method. There is no record of any observation of an emergence of Io on 16 November, but an emergence was observed on 9 November. With this experimental evidence in hand, Rømer explained his new method of calculation to the Royal Academy of Sciences on 22 November.
The original record of the meeting of the Royal Academy of Sciences has been lost, but Rømer's presentation was recorded as a news report in the Journal des sçavans on 7 December. This anonymous report was translated into English and published in Philosophical Transactions of the Royal Society in London on 25 July 1677.[note 5]
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Order of magnitude
Rømer starts with an order of magnitude demonstration that the speed of light must be so great that it takes much less than one second to travel a distance equal to Earth's diameter.
The point L on the diagram represents the second quadrature of Jupiter, when the angle between Jupiter and the Sun (as seen from Earth) is 90°.[note 6] Rømer assumes that an observer could see an emergence of Io at the second quadrature (L), and the emergence which occurs after one orbit of Io around Jupiter (when the Earth is taken to be at point K, the diagram not being to scale), that is 42½ hours later. During those 42½ hours, the Earth has moved farther away from Jupiter by the distance LK: this, according to Rømer, is 210 times the Earth's diameter.[note 7] If light travelled at a speed of one Earth-diameter per second, it would take 3½ minutes to travel the distance LK. And if the period of Io's orbit around Jupiter were taken as the time difference between the emergence at L and the emergence at K, the value would be 3½ minutes longer than the true value.
Rømer then applies the same logic to observations around the first quadrature (point G), when Earth is moving towards Jupiter. The time difference between an immersion seen from point F and the next immersion seen from point G should be 3½ minutes shorter than the true orbital period of Io. Hence, there should be a difference of about 7 minutes between the periods of Io measured at the first quadrature and those measured at the second quadrature. In practice, no difference is observed at all, from which Rømer concludes that the speed of light must be very much greater than one Earth-diameter per second.
However Rømer also realised that any effect of the finite speed of light would add up over a long series of observations, and it is this cumulative effect that he announced to the Royal Academy of Sciences in Paris. The effect can be illustrated with Rømer's observations from spring 1672.
Jupiter was in opposition on 2 March 1672: the first observations of emergences were on 7 March (at 07:58:25) and 14 March (at 09:52:30). Between the two observations, Io had completed four orbits of Jupiter, giving an orbital period of 42 hours 28 minutes 31¼ seconds.
The last emergence observed in the series was on 29 April (at 10:30:06). By this time, Io had completed thirty orbits around Jupiter since 7 March: the apparent orbital period is 42 hours 29 minutes 3 seconds. The difference seems tiny – 32 seconds – but it meant that the emergence on 29 April was occurring a quarter-hour after it would have been predicted. The only alternative explanation was that the observations on 7 and 14 March were wrong by two minutes.
Rømer never published the formal description of his method, possibly because of the opposition of Cassini and Picard to his ideas (see below).[note 8] However, the general nature of his calculation can be inferred from the news report in the Journal des sçavans and from Cassini's announcement on 22 August 1676.
Cassini announced that the new tables would
contain the inequality of the days or the true motion of the Sun [i.e. the inequality due to the eccentricity of the Earth’s orbit], the eccentric motion of Jupiter [i.e. the inequality due to the eccentricity of the orbit of Jupiter] and this new, not previously detected, inequality [i.e. due to the finite speed of light].
Hence Cassini and Rømer appear to have been calculating the times of each eclipse based on the approximation of circular orbits, and then applying three successive corrections to estimate the time that the eclipse would be observed in Paris.
The three "inequalities" (or irregularities) listed by Cassini were not the only ones known, but they were the ones that could be corrected for by calculation. The orbit of Io is also slightly irregular because of orbital resonance with Europa and Ganymede, two of the other Galilean moons of Jupiter, but this would not be fully explained for another century. The only solution available to Cassini and to other astronomers of his time was to issue periodic corrections to the tables of eclipses of Io to take account of its irregular orbital motion: periodically resetting the clock, as it were. The obvious time to reset the clock was just after the opposition of Jupiter to the Sun, when Jupiter is at its closest to Earth and so most easily observable.
The opposition of Jupiter to the Sun occurred on or around 8 July 1676. Rømer's aide-mémoire lists two observation of emergences of Io after this opposition but before Cassini's announcement: on 7 August at 09:44:50 and on 14 August at 11:45:55. With these data, and knowing the orbital period of Io, Cassini could calculate the times of each of the eclipses over the next four to five months.
The next step in applying Rømer's correction would be to calculate the position of Earth and Jupiter in their orbits for each of the eclipses. This sort of coordinate transformation was commonplace in preparing tables of positions of the planets for both astronomy and astrology: it is equivalent to finding each of the positions L (or K) for the various eclipses which might be observable.
Finally, the distance between Earth and Jupiter can be calculated using standard trigonometry, in particular the law of cosines, knowing two sides (distance between the Sun and Earth; distance between the Sun and Jupiter) and one angle (the angle between Jupiter and Earth as formed at the Sun) of a triangle. The distance from the Sun to Earth was not well known at the time, but taking it as a fixed value a, the distance from the Sun to Jupiter can be calculated as some multiple of a.
This model left just one adjustable parameter – the time taken for light to travel a distance equal to a, the radius of Earth's orbit. Rømer had about thirty observations of eclipses of Io from 1671–73 that he used to find the value which fitted best: eleven minutes. With that value, he could calculate the extra time it would take light to reach Earth from Jupiter in November 1676 compared to August 1676: about ten minutes.
Rømer's explanation of the difference between predicted and observed timings of Io's eclipses was widely, but far from universally, accepted. Huygens was an early supporter, especially as it supported his ideas about refraction, and wrote to the French Controller-General of Finances Jean-Baptiste Colbert in Rømer's defence. However Cassini, Rømer's superior at the Royal Observatory, was an early and tenacious opponent of Rømer's ideas, and it seems that Picard, Rømer's mentor, shared many of Cassini's doubts.
Cassini's practical objections stimulated much debate at the Royal Academy of Sciences (with Huygens participating by letter from London). Cassini noted that the other three Galilean moons did not seem to show the same effect as seen for Io, and that there were other irregularities which could not be explained by Rømer's theory. Rømer replied that it was much more difficult to accurately observe the eclipses of the other moons, and that the unexplained effects were much smaller (for Io) than the effect of the speed of light: however, he admitted to Huygens that the unexplained "irregularities" in the other satellites were larger than the effect of the speed of light. The dispute had something of a philosophical note: Rømer claimed that he had discovered a simple solution to an important practical problem, while Cassini rejected the theory as flawed as it could not explain all the observations.[note 9] Cassini was forced to include "empirical corrections" in his 1693 tables of eclipses, but never accepted the theoretical basis: indeed, he chose different correction values for the different moons of Jupiter, in direct contradiction with Rømer's theory.
Rømer's ideas received a much warmer reception in England. Although Robert Hooke (1635–1703) dismissed the supposed speed of light as so large as to be virtually instantaneous, the Astronomer Royal John Flamsteed (1646–1719) accepted Rømer's hypothesis in his ephemerides of eclipses of Io. Edmond Halley (1656–1742), a future Astronomer Royal, was also an early and enthusiastic supporter. Isaac Newton (1643–1727) also accepted Rømer's idea; in his 1704 book Opticks giving a value of "seven or eight minutes" for light to travel from the Sun to Earth, closer to the true value (8 minutes 19 seconds) than Rømer's initial estimate of 11 minutes. Newton also notes that Rømer's observations had been confirmed by others, presumably by Flamsteed and Halley in Greenwich at the very least.
While it was difficult for many (such as Hooke) to conceive of the enormous speed of light, acceptance of Rømer's idea suffered a second handicap in that they were based on Kepler's model of the planets orbiting the Sun in elliptical orbits. While Kepler's model had widespread acceptance by the late seventeenth century, it was still considered sufficiently controversial for Newton to spend several pages discussing the observational evidence in favour in his Philosophiæ Naturalis Principia Mathematica (1687).
Rømer's view that the velocity of light was finite was not fully accepted until measurements of stellar aberration were made in 1727 by James Bradley (1693–1762). Bradley, who would be Halley's successor as Astronomer Royal, calculated a value of 8 minutes 13 seconds for light to travel from the Sun to Earth. Ironically, stellar aberration had first been observed by Cassini and (independently) by Picard in 1671, but neither astronomer was able to give an explanation for the phenomenon. Bradley's work also laid to rest any remaining serious objections to the Keplerian model of the Solar System.
Swedish astronomer Pehr Wilhelm Wargentin (1717–83) used Rømer's method in the preparation of his ephemerides of Jupiter's moons (1746), as did Giovanni Domenico Maraldi working in Paris. The remaining irregularities in the orbits of the Galilean moons would not be satisfactorily explained until the work of Joseph Louis Lagrange (1736–1813) and Pierre-Simon Laplace (1749–1827) on orbital resonance.
In 1809, again making use of observations of Io, but this time with the benefit of more than a century of increasingly precise observations, the astronomer Jean Baptiste Joseph Delambre (1749–1822) reported the time for light to travel from the Sun to the Earth as 8 minutes 12 seconds. Depending on the value assumed for the astronomical unit, this yields the speed of light as just a little more than 300,000 kilometres per second.
The first measurements of the speed of light using completely terrestrial apparatus were published in 1849 by Hippolyte Fizeau (1819–96). Compared to values accepted today, Fizeau's result (about 313,000 kilometres per second) was too high, and less accurate than those obtained by Rømer's method. It would be another thirty years before A. A. Michelson in the United States published his more precise results (299,910±50 km/s) and Simon Newcomb confirmed the agreement with astronomical measurements, almost exactly two centuries after Rømer's announcement.
Did Rømer measure the speed of light?
Several discussions have suggested that Rømer should not be credited with the measurement of the speed of light, as he never gave a value in Earth-based units. These authors credit Huygens with the first calculation of the speed of light.
However, Huygens's estimate was not a precise calculation but rather an illustration at an order of magnitude level. The relevant passage from Treatise sur la lumière reads:
If one considers the vast size of the diameter KL, which according to me is some 24 thousand diameters of the Earth, one will acknowledge the extreme velocity of Light. For, supposing that KL is no more than 22 thousand of these diameters, it appears that being traversed in 22 minutes this makes the speed a thousand diameters in one minute, that is 16-2/3 diameters in one second or in one beat of the pulse, which makes more than 11 hundred times a hundred thousand toises;
Huygens was obviously not concerned about the 9% difference between his preferred value for the distance from the Sun to Earth and the one he uses in his calculation. Nor was there any doubt in Huygens's mind as to Rømer's achievement, as he wrote to Colbert (emphasis added):
I have seen recently, with much pleasure, the beautiful discovery of Mr. Romer, to demonstrate that light takes time in propagating, and even to measure this time;
Neither Newton nor Bradley bothered to calculate the speed of light in Earth-based units. The next recorded calculation was probably made by Fontenelle: claiming to work from Rømer's results, the historical account of Rømer's work written some time after 1707 gives a value of 48203 leagues per second. This is 16.826 Earth-diameters (214,636 km) per second.
It has also been suggested that Rømer was measuring a Doppler effect. The original effect discovered by Christian Doppler 166 years later refers to propagating electromagnetic waves. The generalization referred to here is the change in observed frequency of an oscillator (in this case, Io orbiting around Jupiter) when the observer (in this case, on Earth's surface) is moving: the frequency is higher when the observer is moving towards the oscillator and lower when the observer is moving away from the oscillator. This apparently anachronistic analysis implies that Rømer was measuring the ratio c⁄v, where c is the speed of light and v is the Earth's orbital velocity (strictly, the component of the Earth's orbital velocity parallel to the Earth–Jupiter vector), and indicates that the major inaccuracy of Rømer's calculations was his poor knowledge of the orbit of Jupiter.[note 7]
There is no evidence that Rømer thought that he was measuring c⁄v: he gives his result as the time of 22 minutes for light to travel a distance equal to the diameter of Earth's orbit or, equivalently, 11 minutes for light to travel from the Sun to Earth. It can be readily shown that the two measurements are equivalent: if we give τ as the time taken for light to cross the radius of an orbit (e.g. from the Sun to Earth) and P as the orbital period (the time for one complete rotation), then[note 11]
- Longitude prize (UK)
- Bobis, Laurence; Lequeux, James (2008), "Cassini, Rømer and the velocity of light" (PDF), J. Astron. Hist. Herit., 11 (2): 97–105, Bibcode:2008JAHH...11...97B.
- Bradley, James (1729), "Account of a new discovered Motion of the Fix'd Stars", Philosophical Transactions of the Royal Society of London, 35: 637–60, doi:10.1098/rstl.1727.0064 CS1 maint: discouraged parameter (link).
- Cohen, I. B. (1940), "Roemer and the first determination of the velocity of light (1676)", Isis, 31 (2): 327–79, doi:10.1086/347594, hdl:2027/uc1.b4375710 CS1 maint: discouraged parameter (link); reprinted in book form by the Burndy Library, 1942.
- Daukantas, Patricia (July 2009), "Ole Rømer and the Speed of Light", Optics & Photonics News, 20 (7): 42–47, doi:10.1364/OPN.20.7.000042.
- French, A. P. (1990), "Roemer: a cautionary tale", in Roche, John (ed.), Physicists look back: studies in the history of physics, CRC Press, pp. 120–23, ISBN 0-85274-001-8 CS1 maint: discouraged parameter (link).
- Godin, Louis; Fontenelle, Bernard de, eds. (1729–34), Mémoires de l'Académie royale des sciences depuis 1666 jusqu'en 1692, Paris: Compagnie des libraires, pp. 112–15, 140–41. (in French)
- Huygens, Christiaan (16 September 1677), "Lettre Nº 2103", in Bosscha, J. (ed.), Œuvres complètes de Christiaan Huygens (1888–1950). Tome VIII: Correspondance 1676–1684, The Hague: Martinus Nijhoff (published 1899), pp. 30–31 CS1 maint: discouraged parameter (link). (in Latin)
- Huygens, Christiaan (14 October 1677), "Lettre Nº 2105", in Bosscha, J. (ed.), Œuvres complètes de Christiaan Huygens (1888–1950). Tome VIII: Correspondence 1676–1684, The Hague: Martinus Nijhoff (published 1899), pp. 36–37 CS1 maint: discouraged parameter (link). (in French)
- Huygens, Christiaan (1690), Traitée de la Lumière, Leiden: Pierre van der Aa CS1 maint: discouraged parameter (link). (in French)
- Meyer, Kirstine (1915), "Om Ole Rømers Opdagelse af Lysets Tøven", Det Kongelige Danske Videnskabernes Selskabs Skrifter, 7. Række, Naturvidenskabelig og Mathematisk Afdeling, XII: 3 CS1 maint: discouraged parameter (link). (in Danish)
- Newton, Isaac (1704), "Book II, Prop. XI", Opticks, London: Sam. Smith. and Benj. Walford CS1 maint: discouraged parameter (link).
- Rømer, Ole (30 September 1677), "Lettre Nº 2104", in Bosscha, J. (ed.), Œuvres complètes de Christiaan Huygens (1888–1950). Tome VIII: Correspondance 1676–1684, The Hague: Martinus Nijhoff (published 1899), pp. 32–35 CS1 maint: discouraged parameter (link). (in Latin)
- Saito, Yoshio (2005), "A Discussion of Roemer's Discovery concerning the Speed of Light", AAPPS Bulletin, 15 (3): 9–17.
- Shea, James H. (1998), "Ole Rømer, the speed of light, the apparent period of Io, the Doppler effect, and the dynamics of Earth and Jupiter", Am. J. Phys., 66 (7): 561–69, Bibcode:1998AmJPh..66..561S, doi:10.1119/1.19020.
- Teuber, Jan (2004), "Ole Rømer og den bevægede Jord - en dansk førsteplads?", in Friedrichsen, Per; Henningsen, Ole; Olsen, Olaf; Thykier, Claus; Tortzen, Chr. Gorm (eds.), Ole Rømer - videnskabsmand og samfundstjener, Copenhagen: Gads Forlag, p. 218, ISBN 87-12-04139-4. (in Danish)
- Wróblewski, Andrzej (1985), "de Mora Luminis: A spectacle in two acts with a prologue and an epilogue", Am. J. Phys., 53 (7): 620–30, Bibcode:1985AmJPh..53..620W, doi:10.1119/1.14270.
- There are several alternative spellings of Rømer's surname: Roemer, Rœmer, Römer etc. The Danish Ole is sometimes latinized to Olaus.
- The timing of the emergence comes from one of the few surviving manuscripts of Rømer, in which he records the date as 19 March 1671: see Meyer (1915). By consistency with the other timings recorded in the manuscript (written several years after the event), it has been assumed that Rømer noted the Paris time of the emergence. The time difference of 42 minutes and 10 seconds between Paris and Uraniborg comes from the same manuscript: the value accepted today is 41 minutes 26 seconds.
- Several texts erroneously place the date of the announcement in 1685 or even in 1684. Bobis and Lequeux (2008) have convincingly demonstrated that the announcement was made on 22 August 1676, and that it was made by Cassini and not Rømer.
- The original record of the meeting of the Royal Academy of Sciences has been lost. The quotation comes from an unpublished manuscript in Latin preserved in the library of the Paris Observatory, probably written by Joseph-Nicolas Delisle (1688–1768) at some point before 1738. See Bobis and Lequeux (2008), which contains a facsimile of the manuscript.
- Bobis and Lequeux (2008) tentatively attribute the translation to Edmond Halley (1656–1742), who would become English Astronomer Royal and who is best known for his calculations concerning Halley's comet. However, other sources – not least his own Catalogus Stellarum Australium published in 1679 – suggest that Halley was on the island of St. Helena in the South Atlantic Ocean at the time.
- Although the news report doesn't make it explicit, the choice of a point of quadrature for the example is unlikely to be fortuitous. At the second quadrature, the motion of the Earth in its orbit is taking it directly away from Jupiter. As such, it is the point at which the greatest effect is expected over a single orbit of Io.
- The figure of 210 Earth-diameters per orbit of Io for the orbital speed of the Earth relative to Jupiter is far lower than the real figure, which averages around 322 Earth-diameters per orbit of Io taking into account the orbital motion of Jupiter. Rømer appears to have believed that Jupiter is closer to the Sun (and hence moving faster along its orbit) than is really the case.
- The Royal Academy of Sciences had instructed Rømer to publish a joint paper with his colleagues.
- This last point is put quite clearly as late as 1707 by Cassini's nephew, Giacomo Filippo Maraldi (1665–1729), who also worked at the Royal Observatory: "In order for an hypothesis to be accepted, it is not enough that it agrees with some observations, it must also be consistent with the other phenomena." Quoted in Bobis and Lequeux (2008).
- The exact ratio is 1 toise = 54000⁄27706 metres, or approximately 1.949 m: French law of 19 frimaire An VIII (10 December 1799). Huygens was using Picard's value (1669) of the circumference of the Earth as 360×25×2282 toises, while the 1799 legal conversion uses the more precise results of Delambre and Méchain.
- The expression is given for the approximation to a circular orbit. The derivation is as follows:
(1) express the orbital velocity in terms of the orbital radius r and the orbital period P: v = 2πr⁄P
(2) substitute τ = r⁄c → v = 2πτc⁄P
(3) rearrange to find c⁄v.
- Meyer (1915).
- Rømer (1677).
- Bobis and Lequeux (2008).
- Teuber (2004).
- "Démonstration touchant le mouvement de la lumière trouvé par M. Römer de l'Académie Royale des Sciences" (PDF), Journal des Sçavans: 233–36, 1676. (in French)
- "A demonstration concerning the motion of light, communicated from Paris, in the Journal des Scavans, and here made English", Philosophical Transactions of the Royal Society of London, 12 (136): 893–94, 1677, Bibcode:1677RSPT...12..893., doi:10.1098/rstl.1677.0024, JSTOR 101779
- Saito (2005).
- Huygens (14 October 1677). "J'ay veu depuis peu avec bien de la joye la belle invention qu'a trouvé le Sr. Romer, pour demonstrer que la lumiere en se repandant emploie du temps, et mesme pour mesurer ce temps, qui est une decouverte fort importante et a la confirmation de la quelle l'observatoire Royal s'emploiera dignement. Pour moy cette demonstration m'a agrée d'autant plus, que dans ce que j'escris de la Dioptrique j'ay supposé la mesme chose…"
- Rømer (1677). "Dominos Cassinum et Picardum quod attinet, quorum judicium de illa re cognoscere desideras, hic quidem plane mecum sentit."
- See note 2 at Huygens (16 September 1677).
- In his 1680 Lectures on Light: "so exceedingly swift that 'tis beyond Imagination […] and if so, why it may not be as well instantaneous I know no reason." Quoted in Daukantas (2009).
- Daukantas (2009).
- Newton (1704): "Light is propagated from luminous Bodies in time and spends about seven or eight minutes of an hour in passing from the Sun to the Earth. This was observed first by Romer, and then by others, by means of the Eclipses of the Satellites of Jupiter."
- Bradley (1729).
- Cohen (1940). Wróblewski (1985).
- French (1990), pp. 120–21.
- Huygens (1690), pp. 8–9. Translation by Silvanus P. Thompson.
- Godin and Fonetenelle (1729–34). "Il suit des Observations de Mr. Roëmer, que la lumiére dans une seconde de tems fait 48203 lieuës communes de France, & 377⁄1141 parties d'une de ces lieuës, fraction qui doit bien être négligée."
- Shea (1998).
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