The Moon That Seemed to Run Late

On 9 November 1676, astronomers in Paris watched a moon emerge from Jupiter’s shadow. According to the usual calculation, it should already have appeared. Instead, it arrived about ten minutes late.¹

The moon itself was not late. The delay was in the light carrying news of it.

That distinction—between an event and the moment we see the event—gave astronomy its first persuasive measurement of light’s travel time.

A clock for the whole Earth

The moon was Io, the innermost of Jupiter’s four large satellites. It circles Jupiter once every 1.769 Earth days—about 42 hours and 28 minutes—and, as seen from Earth, is eclipsed by Jupiter once per orbit.²

To a seventeenth-century astronomer, those eclipses looked like the ticks of a distant clock. More importantly, each tick happened at one definite moment for the entire Earth.

The old assumption

The event and the news arrive together

0day of 399

Sun

Earth

Jupiter

Io

Replay 399-day cycle

4.20 AUEarth–Jupiter distance

0:00assumed delay

That made Io potentially useful for finding longitude.

Latitude is comparatively easy to measure: the height of the Sun or a familiar star tells you how far north or south you are. Longitude is harder. The Earth offers no obvious mark in the sky saying how far east or west you have travelled.

Latitude

45° N

Why altitude equals latitude

Latitude is the angle between the observer’s radius and the equatorial plane. Polaris lies almost along Earth’s rotational axis, and its incoming rays are effectively parallel to that axis. The equatorial plane is perpendicular to the axis, so it is also perpendicular to the incoming rays. Meanwhile, the local horizon is perpendicular to the observer’s radius. Thus each side of the latitude angle has been replaced by a perpendicular line: the radius by the horizon, and the equator by the Polaris ray. Rotating both lines through 90° preserves the angle between them, so latitude equals the altitude of Polaris. In practice the equality is approximate because Polaris is currently about 0.65° from the true celestial pole.

But the Earth rotates at a known rate: 15 degrees each hour. If it is midnight where you are when a predicted eclipse of Io occurs at 10 p.m. Paris time, your local time differs from Paris by two hours, and your longitude differs by 30 degrees.

The method was too awkward for a rolling ship. A telescope powerful enough to see Io was difficult to aim from a moving deck, and Jupiter disappeared into the Sun’s glare for part of every year. On land, however, the Jovian moons became valuable tools for surveying and mapmaking.³

The hard part was predicting their eclipses accurately.

At the Paris Observatory, Giovanni Domenico Cassini directed a long campaign to improve those predictions. Ole Rømer, a young Danish astronomer working with Cassini and Jean Picard, helped observe the moons and study the irregularities in their tables.

The work was collective. Cassini had recognized that the eclipse times contained systematic discrepancies, and surviving Academy records show that in August 1676 he connected one of them with the changing distance between Earth and Jupiter—and even suggested that light might take time to cross that distance. Rømer became the idea’s clearest advocate. He worked out the argument, defended it against competing explanations, and presented it to the Academy that November.⁴

A tiny delay that adds up

Io’s orbit is not perfectly uniform. Europa and Ganymede perturb it, and seventeenth-century astronomers did not yet have a complete theory of those interactions. The historical timings also contained clock errors, transcription mistakes and ordinary observational scatter.⁵

So a single eclipse arriving slightly early or late proved very little.

Rømer’s insight concerned what happened over many eclipses.

Suppose Io emerges from Jupiter’s shadow at a time we can call $t_0$. One orbit later it emerges again, after a true interval $P$:

$$t_1=t_0+P.$$

But that is not quite the interval measured on Earth. We see each event only after its light has crossed the Earth–Jupiter distance. If that distance is $D$, the arrival time is delayed by $D/c$, where $c$ is the speed of light:

$$t_{\mathrm{seen}}=t_{\mathrm{event}}+\frac{D}{c}.$$

While Earth is moving away from Jupiter, the second signal has farther to travel than the first. The observed interval is therefore

$$P+\frac{\Delta D}{c}.$$

While Earth is approaching Jupiter, the interval is instead slightly shorter than $P$.

The difference for one orbit of Io is tiny. During one 42.5-hour circuit of Io, Earth travels about 4.6 million kilometres along its orbit. Only the component of that motion along the Earth–Jupiter line affects the signal’s travel time. Near the geometry where that component is largest, the resulting timing change is of order fifteen seconds per orbit.⁶

Fifteen seconds was difficult to establish with one seventeenth-century observation. But after forty revolutions, fifteen seconds per revolution becomes ten minutes. The small delay accumulates until it is larger than the uncertainty of any one eclipse.

This is the heart of the method. Rømer did not need a flawless observation. He needed a small error that repeatedly changed in the same direction.

The messenger, not the moon

There were two broad explanations for the accumulating discrepancy.

Perhaps Io was genuinely changing its pace. That was not absurd: the Jovian moons do perturb one another, and Cassini had good reason to take orbital irregularities seriously.

But the particular discrepancy followed the motion of the observer. Io’s eclipses appeared progressively delayed while Earth receded from Jupiter and progressively advanced while Earth approached.

For Io itself to behave that way, a moon hundreds of millions of kilometres away would somehow have to adjust its orbit according to Earth’s position. The more economical explanation was that nothing at Jupiter had changed. What changed was the journey from Jupiter to us.

Imagine a perfectly punctual bell rung every hour in a distant town. A cyclist leaves the town at each ring to bring you the news. If you are travelling away, each cyclist has farther to ride than the previous one, so the reports reach you at intervals slightly longer than an hour. If you are travelling toward the town, they arrive at intervals slightly shorter than an hour.

The bell has not changed its rhythm. The travel time of the message has changed.

Io was the bell. Light was the messenger.

Why the pattern takes 399 days

The geometry repeats not every Earth year, but every time Earth catches Jupiter again.

Earth circles the Sun in about 365 days. Jupiter requires nearly twelve years. Because Earth runs on the inside track, it laps Jupiter once every 399 days. Astronomers call this the Earth–Jupiter synodic period.

During that cycle, the Earth–Jupiter distance first decreases and then increases. The light-travel delay changes with it. This is why the discrepancy follows neither Io’s 42.5-hour orbit nor exactly the 365-day calendar. It follows the changing relationship between two moving planets.

To first approximation, the full near-to-far change in distance is the diameter of Earth’s orbit: two astronomical units. With modern definitions, light crosses that distance in about 998 seconds, or 16 minutes and 38 seconds.⁷

What Rømer realised

The news has to cross the distance

0day of 399

Sun

Earth

Jupiter

Io

Replay 399-day cycle

4.20 AUEarth–Jupiter distance

34:56light-travel time

Rømer’s estimate was about 22 minutes. It was not especially accurate, but it was finite and of the right order. The uncertainty came from noisy eclipse timings, genuine irregularities in Io’s motion, and imperfect seventeenth-century models of the planetary orbits.

The important achievement was not the exact number. It was finding a method that would give a better answer when supplied with better observations.

The prediction

In August and early September 1676, the Paris astronomers committed themselves in advance. An eclipse later that year, they said, would appear roughly ten minutes later than a prediction extrapolated from observations made when Earth was closer to Jupiter.

The surviving historical record is untidy. An Academy note attributes a forecast for 16 November to Cassini; the report published that December credits Rømer with predicting the effect and records an eclipse observed on 9 November at 5:35:45 p.m., about ten minutes late.⁸

What matters scientifically is that the delay was predicted before the confirming observation. The finite-speed explanation did not merely accommodate old discrepancies. It successfully said what observers should see next.

On 21 November, Rømer presented the argument to the Académie Royale des Sciences. A short account appeared in the Journal des Sçavans on 7 December. Not everyone accepted it. Cassini soon withdrew from the finite-light-speed interpretation because comparable delays could not yet be established cleanly for Jupiter’s other moons. Some astronomers continued to prefer unknown orbital irregularities.

But the idea gained influential supporters, including Christiaan Huygens, Isaac Newton, Edmond Halley and John Flamsteed. Half a century later, James Bradley’s discovery of stellar aberration supplied independent evidence that light’s speed was finite.⁹

A time before a speed

Rømer did not publish a velocity in kilometres or miles per second. His result was a travel time: about 22 minutes across the diameter of Earth’s orbit.

That was already a measurement of light’s propagation. To convert a travel time into a speed, one also needed an estimate of the diameter of Earth’s orbit, which was still imprecise.

Huygens performed the conversion in a work written in 1678. Modern reconstructions place his answer at roughly $2.2\times10^5$ kilometres per second, compared with the modern value of 299,792 kilometres per second.¹⁰

The numerical estimate was rough. The conceptual result was revolutionary: when we look into space, we do not see events as they are happening. We see messages sent in the past.

Looking at Jupiter now

Steady binoculars can often show three or four Galilean moons as points of light beside Jupiter. They are the same bodies Galileo saw in 1610 and the same clockwork Cassini and Rømer studied in Paris.

Depending on the planets’ positions, light from Jupiter takes approximately 33 to 54 minutes to reach Earth. An eclipse seen through a telescope is therefore already over before its image enters the eyepiece.¹¹

Nothing is wrong with the moon. Nothing has been delayed at Jupiter.

The event happened on time. The messenger had a long way to travel.

Sources and notes

1. Ole Rømer, “A Demonstration Concerning the Motion of Light”, Philosophical Transactions 12 (1677), pp. 893–894. This contemporary report gives the observation time, the roughly ten-minute delay and Rømer’s prediction.

   ↩

2. American Museum of Natural History, “Ole Rømer and the Speed of Light”. It identifies Io as the innermost of Jupiter’s four large satellites, gives its period as 1.769 days and says it is eclipsed once per orbit.

   ↩

3. Museo Galileo, “Galilean Method for Determining Longitude”; The Galileo Project, Rice University, “Longitude at Sea”; Royal Observatory Greenwich, “The Quest for Longitude and the Rise of Greenwich”.

   ↩

4. Laurence Bobis and James Lequeux, “Cassini, Rømer and the Velocity of Light”, Journal of Astronomical History and Heritage. The surviving record makes priority more complicated than the familiar Rømer-only account.

   ↩

5. NASA, “Io Facts”, for perturbations by Europa and Ganymede; “Rømer’s determination of the speed of light”, for the historically incomplete theory of Io’s orbital resonance; and K. S. Kristensen and K. M. Pedersen, “Roemer, Jupiter’s Satellites and the Velocity of Light”, for clock corrections, timing errors and erroneous entries in the surviving observations.

   ↩

6. American Museum of Natural History, “Ole Rømer and the Speed of Light”, gives Io’s period as 1.769 days; NASA, “The Solar System”, gives Earth’s mean orbital speed as 29.78 km/s; and the International Bureau of Weights and Measures, “The SI”, defines $c$ as 299,792,458 m/s. Thus Earth travels about $1.769\times86{,}400\times29.78\approx4.55$ million km during one Io orbit, corresponding to an upper scale of about 15 seconds of light-travel time.

   ↩

7. American Museum of Natural History, “Ole Rømer and the Speed of Light”, describes the additional path as the diameter of Earth’s orbit. The International Bureau of Weights and Measures’ SI Brochure gives one astronomical unit as exactly 149,597,870,700 metres, while its “The SI” page defines $c$ as 299,792,458 m/s. Therefore $2\,\mathrm{au}/c=998.01$ seconds.

   ↩

8. Rømer, “A Demonstration Concerning the Motion of Light”; Bobis and Lequeux, “Cassini, Rømer and the Velocity of Light”.

   ↩

9. Bobis and Lequeux, “Cassini, Rømer and the Velocity of Light”; James Bradley, “A New Discovered Motion of the Fix’d Stars”, Philosophical Transactions 35 (1729).

   ↩

10. Christiaan Huygens, Treatise on Light; Laurence Bobis and James Lequeux, “Cassini, Rømer and the Velocity of Light”, which gives approximately 230,000 km/s; and American Museum of Natural History, “Ole Rømer and the Speed of Light”, which gives 131,000 mi/s, approximately 211,000 km/s. The difference reflects alternative treatments of historical units and assumptions.

    ↩

11. NASA, “Jupiter to Reach Opposition, Closest Approach to Earth in 59 Years”, says that good, steady binoculars can reveal three or four Galilean moons. NASA’s “Jupiter Fact Sheet” gives minimum and maximum Earth–Jupiter distances of approximately 588.5 and 968.1 million kilometres. Dividing by the International Bureau of Weights and Measures’ exact speed of light gives approximately 32.7 and 53.8 minutes.

    ↩