Ole Rømer

How to use this course

Ole Rømer is remembered for one result: light takes time to travel, and the eclipses of Jupiter’s moon Io show how much. That result came from a working astronomer whose daily business was precise timing. Rømer built instruments, gearing and calculating machines, standardized weights and measures for a kingdom, reformed its calendar and ran the police of its capital. This course treats the speed of light in detail and places it among the other problems of measurement he solved.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, derivations and exercises with folded solutions. The course on Giovanni Domenico Cassini covers the same events in Paris from Cassini’s side; the two are written to agree.

Timeline

Aarhus and Copenhagen, 1644–1671

Ole Christensen Rømer was born in Aarhus on 25 September 1644. His father, Christen Pedersen, came from the island of Rømø and took the surname Rømer when he settled in Aarhus in 1642. He was a merchant and skipper who owned navigational instruments and books. His mother, Anna Olufsdatter Storm, was the daughter of an Aarhus alderman. The father died in 1663.

Rømer left Aarhus Cathedral School in 1662 and went to the University of Copenhagen, then the only university in Denmark–Norway. There the decisive influence was Rasmus Bartholin, whose discovery of double refraction in crystals of Iceland spar was published while Rømer lived in his house. In 1655 King Frederik III had bought Tycho Brahe’s original observation logs from Johannes Kepler’s son, and in 1664 he ordered Bartholin to publish them. Bartholin took the student Rømer, after two years of study, as his assistant and lodged him in his house.

The editing gave Rømer an education no lecture course could provide. Tycho’s logs were the most accurate naked-eye observations ever made, recorded with notes on instruments, corrections and errors. The edition was abandoned when Frederik III died in 1670 and was printed only in the twentieth century, but Rømer had learned at first hand what precision astronomy demanded.

Hven with Picard, 1671

In 1671 the French Academy of Sciences sent Jean Picard to Denmark. Picard had just measured an arc of the meridian near Paris, and he needed the exact position of Tycho’s observatory of Uraniborg on the island of Hven to compare Tycho’s observations with those made in Paris. Uraniborg had been built around 1580 and torn down by 1601; Picard found the site overgrown and partly used as a dump. Bartholin and Rømer went with him to locate the foundations and set up instruments.

The key measurement was longitude. Picard and Rømer observed eclipses of Jupiter’s innermost satellite, Io, at Hven, while Cassini observed the same events in Paris. The difference of local times gives the difference of longitude (Lesson 04). Over several months they recorded a long series of eclipses; Wikipedia’s account gives the figure of about 140. One of the Hven–Paris comparisons gave a time difference of 42 minutes 10 seconds, within about 45 seconds of the modern value.

Picard was impressed by the young Dane and obtained King Christian V’s permission to take him to Paris. Rømer travelled there in 1672, taking Tycho’s observations with them.

Paris, 1672–1681

The Paris Observatory had opened in 1671 under Giovanni Domenico Cassini. Rømer lived and worked there for nearly nine years and was soon admitted to the Académie royale des sciences. He observed Jupiter’s satellites as part of Cassini’s programme to perfect their tables, and he built instruments, including an improved micrometer.

He also served the court. He taught mathematics to the Dauphin, the heir of Louis XIV, and for teaching designed mechanical models: a planetary machine showing the motions of the six known planets around the Sun, a Jovilabe for Jupiter’s satellites and an eclipse machine (Lesson 08). The models were built in four copies with the help of a Paris clockmaker. One set stayed in Paris, one went to Christian V in Copenhagen and is now at Rosenborg Castle, and two were sent as gifts to the King of Siam and the Emperor of China. Rømer also worked on the hydraulics and levelling of the water supply for the royal fountains at Versailles. In 1674 he proposed the epicycloid as the correct profile for gear teeth (Lesson 07).

The work that made him famous began as a correction to the tables of Io. On 22 August 1676 Cassini announced to the Academy that his eclipse tables would include a new inequality, and mentioned that it could be explained by a finite speed of light. Rømer developed that explanation. He predicted that an emergence of Io in November 1676 would be seen about ten minutes later than the old tables gave, and on 21 November he presented his theory to the Academy: light takes about 22 minutes to cross the diameter of the Earth’s orbit. An anonymous summary appeared in the Journal des sçavans on 7 December 1676, and an English version in the Philosophical Transactions in 1677. Rømer never published a full account himself.

Cassini then turned against the idea. The other satellites did not show the effect clearly, and Io had other irregularities. Christiaan Huygens, by contrast, corresponded with Rømer and in 1677 wrote to the minister Colbert in support. In 1679 Rømer visited England, where he met John Flamsteed, the Astronomer Royal, who began to apply the light-time correction to his own tables of the satellites, and Edmond Halley. In 1681 Christian V recalled him to Copenhagen.

A clock in the sky: longitude from Io

For the general reader. Latitude can be read from the height of the pole star; longitude is a difference of clock times. To find how far east of Paris you stand, you need to know what time it is in Paris at the moment when it is noon where you are. Galileo proposed using the moons of Jupiter as a clock visible from everywhere at once. When a moon enters Jupiter’s shadow, every observer on Earth sees it disappear at the same instant, whatever their local time.

The Earth turns \(360^\circ\) in 24 hours of mean solar time, \(15^\circ\) per hour, and local time is fixed by the Sun’s position relative to the local meridian. An eclipse of a Jovian satellite is, to a very good approximation, seen simultaneously from all points of the Earth. The light time across the Earth is about 0.04 s, negligible here (though the light time across the Earth’s orbit is the subject of Lesson 05).

Periods

The eclipses of Io recur with the synodic period, the time for Io to return to the same position relative to the Sun–Jupiter line, because Jupiter’s shadow points away from the Sun. If Io’s sidereal period is \(P_I\) and Jupiter’s is \(P_J\),

With \(P_I=1.769138\) d and \(P_J=4332.59\) d, \(S\approx1.76986\) d, or 42 h 28.6 min. Near the opposition of Jupiter, eclipses are seen as immersions (disappearances) on one side of the planet before opposition and as emergences after it; near conjunction Jupiter is lost in the Sun’s glare and nothing is seen for weeks.

The delay of light, 1676

For the general reader. If light takes time to travel, an event at Jupiter is seen late, and the delay depends on how far away Jupiter is. As the Earth moves along its orbit away from Jupiter, each eclipse of Io is seen a little later than the one before it should have been; as the Earth approaches, a little earlier. Over months, the small differences add up to many minutes. Rømer saw that this single cause explained the drift of Io’s eclipses, and that the drift measures the time light needs to cross the Earth’s orbit.

Derivation. Differentiate \(t+\Delta(t)/c\) with respect to \(t\): consecutive events separated by \(S\) at the source arrive separated by \(S(1+\dot\Delta/c)\) to first order. Summing gives the second formula. For circular coplanar orbits, \(\Delta\) ranges from \(a_J-a_\oplus\) at opposition to \(a_J+a_\oplus\) at conjunction.

Rømer’s argument

The observable effect is not \(2a_\oplus/c\) directly, since eclipses cannot be seen near conjunction. What one sees is that a period measured over many orbits while the Earth recedes from Jupiter is longer than one measured while it approaches, and that the accumulated difference grows in step with the change of distance. Cassini’s tables were reset at the opposition of 8 July 1676. By November the Earth had moved far around its orbit and the distance to Jupiter had grown by more than an astronomical unit, so emergences should be seen roughly ten minutes later than tables built without the correction. The November observations agreed. Rømer combined his long series of observations, which by his own count exceeded seventy by the end of 1677, into the estimate of about 22 minutes for the diameter of the orbit.

Why Io

Io has the shortest period and the most frequent, sharpest eclipses. The light-time delay is the same for every satellite, but Cassini’s objection had force: the outer satellites have longer periods, so fewer eclipses, and their orbits have perturbations, mostly from their mutual resonances, which Rømer could not separate from the light-time effect. The full theory of the satellites, with mutual perturbations, had to wait for Pierre-Simon Laplace at the end of the eighteenth century.

From light time to the speed of light

For the general reader. Rømer measured a time, not a speed. To turn “22 minutes to cross the Earth’s orbit” into kilometres per second, one needs the size of the orbit, which in 1676 was known only roughly. Huygens made the conversion in 1690. Fifty years after Rømer, James Bradley found a second, independent effect of the finite speed of light: the stars appear shifted in the direction of the Earth’s motion. The two methods agreed, and the finite speed of light was no longer in doubt.

Huygens’s conversion

In his Traité de la lumière (1690), Huygens took Rømer’s 22 minutes and a diameter of the Earth’s orbit of about 22 000 Earth diameters. Light then travels about 1000 Earth diameters per minute, or about \(16\tfrac23\) Earth diameters per second. In modern units this is about 212 000 km/s. The error comes from both inputs: the true light time across the orbit is 16.6 minutes, and the true diameter of the orbit is about 23 500 Earth diameters. In 1704 Isaac Newton gave seven or eight minutes for light to come from the Sun in his Opticks, much closer to the truth.

Aberration

The proof is the rain analogy made exact. In the observer’s frame the light’s velocity is the vector difference of its velocity in the Sun’s frame and the observer’s velocity \(\mathbf v\). Resolving along and across \(\mathbf v\), the apparent direction \(\theta'\) satisfies \(\tan\theta'=\sin\theta/(\cos\theta+v/c)\), so \(\theta-\theta'\approx(v/c)\sin\theta\) to first order. Bradley, working with Samuel Molyneux from 1725 on the star γ Draconis, found an annual displacement he explained in 1728 in this way, and published in 1729. His constant of about \(20.2''\) implied that light takes 8 minutes 12 seconds to reach us from the Sun, in good agreement with eclipse timings.

Aberration confirmed Rømer independently of any theory of Jupiter’s satellites. It also showed that the Earth moves, which was the observational proof of Copernican motion that Rømer himself had sought through stellar parallax (Lesson 09).

Epicycloidal gear teeth

For the general reader. Two gears with badly shaped teeth turn jerkily: the driven wheel speeds up and slows down as each pair of teeth slides past. Clocks and planetary models need smooth, steady motion. In 1674, according to standard accounts, Rømer proposed that teeth be cut to the shape of an epicycloid, the curve traced by a point on a circle rolling around another circle. With such teeth the ratio of speeds stays exactly constant.

The law of gearing

Proof. The profiles stay in contact, so the velocities of the contact point as a point of each body have equal components along the common normal \(\mathbf n\): \(\omega_1\,d_1=\omega_2\,d_2\), where \(d_i\) is the distance from \(O_i\) to the normal line. The triangles formed by \(O_1,O_2\), the normal and \(P\) are similar, so \(d_1/d_2=O_1P/O_2P\).

Cycloidal profiles

Idea. At each instant the generating circle rotates about its point of contact with the pitch circles, which is \(P\). The tracing point therefore moves perpendicular to the line from \(P\) to itself, so the normal to each traced curve passes through \(P\). Both profiles share the tracing point and its normal, so they touch there with a common normal through the fixed point \(P\), and the law of gearing applies.

Cycloidal teeth dominated clockmaking for centuries. Engineering gears later moved to the involute profile, which tolerates errors in the distance between centres, but watch and clock trains still often use cycloidal forms.

Planetary machines and gear ratios

For the general reader. A planetarium driven by a crank must make Saturn go round once while the Earth goes round about 29½ times. Gears with whole numbers of teeth can only produce ratios of whole numbers, and the true ratios are not simple fractions. Every maker of astronomical machines, from Rømer in Paris onwards, had to choose tooth counts that approximate the real periods well enough with gears small enough to cut.

Best approximations

Every real \(x>0\) has a continued fraction \(x=a_0+\cfrac{1}{a_1+\cfrac{1}{a_2+\cdots}}\), written \([a_0;a_1,a_2,\dots]\). Its truncations \(p_n/q_n\) are the convergents, computed by \(p_n=a_np_{n-1}+p_{n-2}\), \(q_n=a_nq_{n-1}+q_{n-2}\).

The first statement follows from the identity \(p_{n+1}q_n-p_nq_{n+1}=(-1)^n\) and the fact that \(x\) lies between consecutive convergents. The second is Lagrange’s theorem on best approximations. For a gear train, \(q\) and \(p\) are tooth counts or products of tooth counts, so the convergents tell the instrument maker which small gears are worth using.

The Jovilabe

The sidereal periods of Io, Europa and Ganymede are 1.769, 3.551 and 7.155 days, very nearly in ratio \(1:2:4\). Laplace later showed that the mean motions satisfy \(n_I-3n_E+2n_G=0\) exactly, so the three moons can never be in line on the same side of Jupiter. A Jovilabe geared in the simple ratios \(1:2:4\) reproduces the satellites’ configurations well for some days; any real instrument must be reset from observations.

The transit instrument and meridian circle

For the general reader. As the Earth turns, every star crosses the local meridian, the north–south line through the zenith, once a day. If a telescope can swing only in the plane of the meridian, then the clock time at which a star crosses the crosshair gives its east–west coordinate, and the angle of the telescope gives its north–south coordinate. Rømer built such a transit instrument around 1690 and added a precisely divided circle in 1704. This design became the standard tool of positional astronomy for two hundred years.

Sidereal time is defined as the hour angle of the vernal equinox, and a star’s hour angle is \(\mathrm{LST}-\alpha\). At transit the hour angle is zero. For the altitudes, the pole stands at altitude \(\phi\) above the north horizon, and a star circles it at angular distance \(90^\circ-\delta\). Refraction raises both culminations slightly, by different amounts, and must be corrected.

Rømer’s instrument turned on a horizontal east–west axis supported by two piers, so the telescope could only sweep the meridian. Compared with the large quadrants and sextants of Tycho, it reduced the measurement of one coordinate to timing with a good pendulum clock, which, since Huygens, could keep time to a few seconds a day. Rømer hoped to detect the parallax of the stars and so prove the Earth’s motion. The Triduum, three nights of observations on 20–23 October 1706, gave positions of 88 stars from about 250 observations made at the Tusculanum.

Two fixed points: the Rømer thermometer

For the general reader. Seventeenth-century thermometers had arbitrary marks, so readings from two instruments could not be compared. Around 1702 Rømer made alcohol thermometers calibrated at two reproducible temperatures, the freezing and the boiling of water, with the degrees in between spaced equally. Daniel Gabriel Fahrenheit visited him in 1708, learned the method, and built on it the scale that still bears Fahrenheit’s name.

The scale is only as good as the assumption that the liquid expands uniformly. Different liquids expand nonlinearly in different ways, so alcohol and mercury thermometers calibrated at the same two points disagree slightly in between. The problem was resolved only with the thermodynamic temperature of the nineteenth century. Rømer’s choice of 7.5 for freezing has no documented explanation; one proposal is that his zero was first set at the freezing point of a salt solution. His notes on the thermometer survive in his notebook, the Adversaria, published in 1910.

Fahrenheit, according to his own later correspondence with Herman Boerhaave, adopted the scale, raised the freezing point from 7.5 to 8 and the mark near body heat from 22.5 to 24, and then divided each degree into four. Freezing became 32 and body heat 96.

Copenhagen, 1681–1710

Rømer returned to Copenhagen in 1681 as professor of astronomy at the university; from 1686 he directed the observatory on top of the Round Tower. The same year he married Anne Marie Bartholin, daughter of his teacher. Over the next three decades he held a remarkable range of offices, and much of his time went to the state.

As royal mathematician he prepared the first national system of weights and measures for Denmark–Norway, introduced on 1 May 1683 and refined in 1698. He defined the Danish mile as 24 000 Danish feet, about 7.5 km. His survey of the main roads led to the first Danish milestones, some of which still stand. He helped establish schools of navigation. In 1700, after discussing the reform with Gottfried Wilhelm Leibniz, he persuaded the king to adopt the reformed calendar: in Denmark–Norway, 18 February 1700 was followed directly by 1 March.

In 1705 he became police director and mayor of Copenhagen. He set building regulations, improved water supply, sewers and fire equipment, introduced oil lamps for street lighting and organized measures on poverty and begging. He also served as a judge in the Supreme Court, as university librarian and twice as rector of the university, and sat on many royal commissions.

He did not give up astronomy. Disturbed by the bells of the church beneath the Round Tower, he built a transit instrument at his house in Store Kannikestræde around 1690, and in 1704 a country observatory, the Tusculanum, at Vridsløsemagle west of the city, on Bartholin family land. Its meridian circle produced the Triduum of October 1706. His assistant Peder Horrebow later published an account of his instruments. Rømer died in Copenhagen on 19 September 1710, six days before his 66th birthday, and was buried in the Church of Our Lady, Copenhagen’s cathedral.

Fire, survival and legacy

On 20 October 1728 a fire broke out in Copenhagen and burned for three days. It destroyed much of the old city, including university buildings, and with them most of Rømer’s observations and instruments. Three things survived. The Triduum survived, of which Rømer had five copies made. His working notebook, the Adversaria, survived and was published in 1910. And Peder Horrebow, who had worked with him, described his instruments and methods in Basis astronomiae (1735), which spread his designs across Europe. A folio listing his eclipse observations of 1668–1677 was found in 1913.

• The speed of light. Rømer’s light time was accepted by Huygens, Flamsteed, Halley and Newton, resisted in Paris, and confirmed beyond doubt by Bradley’s aberration of 1728. Pehr Wargentin’s ephemerides of Jupiter’s satellites (1746) included the light-time correction, and Jean-Baptiste Delambre in 1809 derived 8 minutes 12 seconds for the light time from the Sun from a long series of eclipses. The first terrestrial measurement, by Hippolyte Fizeau in 1849, came 173 years after Rømer’s announcement.
• Instruments. The transit instrument and the meridian circle were the backbone of positional astronomy until photography and, much later, astrometric satellites.
• Measurement and the state. His weights and measures, milestones and calendar reform belong to the same programme as his astronomy: reproducible standards, carefully defined.
• Temperature. Through Fahrenheit, his two-point calibration became standard practice.
• Honours. The Ole Rømer Medal of the Danish research councils, the Ole Rømer Observatory of Aarhus University, the Kroppedal Museum beside the excavated Tusculanum (rediscovered in 1978), the lunar crater Römer, and streets in Aarhus and Copenhagen bear his name.

Sources and further reading

Biography

1. Ole Rømer, Wikipedia.
2. Niels Bohr Institute, Ole Rømer: education, Copenhagen, Machines and Observatorium Tusculanum.
3. Ole Rømer, Den Store Danske (lex.dk).
4. Ole Rømer and the speed of light, Optics and Photonics News 20 (2009).

Physics and astronomy

1. [O. Rømer], Démonstration touchant le mouvement de la lumière, Journal des sçavans, 7 December 1676; English translation in Philosophical Transactions 12 (1677).
2. Rømer’s determination of the speed of light, Wikipedia (detailed chronology and numbers).
3. L. Bobis and J. Lequeux, Cassini, Rømer and the velocity of light, Journal of Astronomical History and Heritage 11 (2008).
4. I. B. Cohen, Roemer and the first determination of the velocity of light (1676), Isis 31 (1940).
5. C. Huygens, Traité de la lumière, Leiden, 1690.
6. J. Bradley, An account of a new discovered motion of the fix’d stars, Philosophical Transactions 35 (1729); Aberration (astronomy), Wikipedia.
7. Rømer scale, Wikipedia.
8. P. Horrebow, Basis astronomiae, Copenhagen, 1735.
9. R. Willis, Principles of Mechanism, 1841 (cycloidal and involute gearing).
10. A. Ya. Khinchin, Continued Fractions, Dover (best approximations).