Talking across the Solar System: how long is the delay?

Pick a body, then slide it around its orbit with the angle slider. One question to Mars, and one answer home. The model gives 8.7 min — or 42.0 min. Which one is the delay?

In 1676 Ole Rømer timed a moon of Jupiter coming out of Jupiter's shadow. The moon was clockwork. The timings were not. They ran early when Earth was near Jupiter. They ran late when Earth was far. Rømer said the shadow was on time and the light was late.

He put a number on it. By his figure, light needed about 22 minutes to cross Earth's orbit, end to end. Many people did not accept that light took any time at all. The argument took decades to close.

There is no single distance to cross. Pick a body below, then move it around its orbit with the angle slider. The one-way figure for Mars moves by nearly five times. Switch between One way and There and back and the big number doubles or halves at every angle, apart from rounding, so that factor stays the same.

Distance sets the delay. Earth's own ring makes it swing.

Ask how long a message takes to reach Mars and you expect one number. The model does not have one. It has a band. The one-way figure runs from 4.4 min to 21.0 min. Quote a figure with no angle and it is loose by 4.82×.

That figure comes from the card marked Closest to farthest. Every body has its own. The model puts Venus at 6.23× and Neptune at 1.07×. Venus is the near planet and Neptune the far one. The near one swings far more.

Earth is the reason. The model measures from Earth's ring, not from the Sun. That ring has a radius of one astronomical unit. So the model takes the same 1.00 au off one end. Then it adds that same step to the other end, every time. Against Venus's ring of 0.72 au that step is most of the answer. Against Neptune's 30.07 au it is a rounding error.

The delay itself runs the other way. The model puts the Sun at 8.3 min. It puts Neptune between 4 h 2 min and 4 h 18 min. So in this model Earth's orbit is what makes the swing, and the target's distance decides how much it matters. The delay columns and the Swing column disagree about which body is interesting, though every figure in them comes from the same eight rows. The bars above draw both — position for the delay, length for the swing — and here is how to read the whole widget.

Mars at both ends, counted by hand

Take Mars at θ = 180° and follow one number down. The two radii add: 1.523 710 34 + 1.000 002 61 = 2.523 712 95 au. In metres that is 2.523 712 95 × 149 597 870 700, or 3.7754 × 10¹¹ m. Call it 377.5 million kilometres. Divide by c.

The card rounds that to 21.0 min. There and back doubles it to 2 518.7 s, or 42.0 min.

The same steps at θ = 0° subtract instead. 1.523 710 34 − 1.000 002 61 = 0.523 707 73 au, or 78.3 million kilometres. Dividing by c gives 261.3 s, which the card shows as 4.4 min.

Now divide the two one-way figures: 1 259.3 ÷ 261.3 = 4.82. That is the swing column, reached from two distances rather than read off the row. The same ratio holds in kilometres, in seconds and in astronomical units. The only step between those units is a division by a constant.

The card under How many times you each speak multiplies from there. The model puts eight turns at θ = 180° at 5 h 36 min. It puts the same eight turns at θ = 0° at 1 h 10 min. Nothing changed but where the two bodies were standing. All of it rests on three rules.

Three rules, and two numbers fixed by definition

Drop any one of these and the figures stop meaning what they say.

1. Each body rides a circle around the Sun. Its radius is that body's orbital semi-major axis. All the circles lie in one plane.
2. The signal runs straight through empty space at exactly c.
3. Nobody takes any time to think. A turn counts travel and nothing else.

There is no simulation. Every figure is the closed form worked once in double precision. So the largest error in the table is the geometry, not the arithmetic.

The two exact numbers. The speed of light in vacuum is 299 792 458 m/s. That is a defined value, not a measured one. The 17th CGPM fixed the metre by it in 1983. The astronomical unit is 149 597 870 700 m, adopted by the IAU in 2012. Neither carries an uncertainty, so 1 au = 499.004784 light-seconds is exact to the digits shown.

The radii. Each one is a semi-major axis, not a mean distance and not a current one. The values are the J2000 elements in JPL's Approximate Positions of the Planets. They are good for 1800–2050. The row used for Earth is the Earth–Moon barycentre. That offset sits far below the error the circles already carry. Circles are the large omission, and the next section prices them in minutes.

Real orbits are ellipses, and that breaks both ends

Eccentricity. Real orbits are ellipses, and circles narrow the band at both ends. From the same elements, Mars's perihelion is 1.3814 au and Earth's aphelion 1.0167 au. The difference is 0.3647 au, or 3.0 min. The two aphelia add to 2.6827 au, or 22.3 min. On this page those ends are 4.4 min and 21.0 min. The three minutes often quoted for Mars is not a figure the circles can produce. Only the ellipses reach it. The widget prints both bands under its table, so the gap stays visible.

One plane. Every orbit lies flat here. A tilt lifts one body off the flat plane. At the same angle measured along the plane, that lengthens the chord near θ = 0° and shortens it near θ = 180°, by under a second for Mars either way.

The Sun in the way. Near θ = 180° the Sun stands between the two bodies. The model still returns a delay, because it holds only geometry. It has no term for solar plasma along the path. It has nothing for signal strength, or for the weeks when commanding stops.

No clock. There is no date input. So the model cannot say when a line-up happens, or how long it lasts. It has no term for relay hops or for atmosphere. It has none for the light-time correction near the Sun. The width of Earth's own orbit is one distance people went out and measured, again and again: by parallax in 1672, by transits of Venus in 1761 and 1769 and again in 1874 and 1882, and by radar from 1961.

Rømer was long by a third. The method held.

Press Sunlight reaching Earth and the card reads 8.3 min. Double it for the full width of Earth's orbit and the model gives 16.6 min. Rømer said about 22 minutes. He was long by roughly a third.

The size of that error is not the interesting part. Rømer had no trustworthy distance to the Sun to work with. He gave his answer as a time. 22 minutes is what that looks like. Turning his time into a speed needed that distance too.

Clear radar echoes from Venus arrived in 1961. Several groups timed them. Three of their papers are listed below. The method ran the same division the other way round. The speed was well known by then, so timing the echo gave the distance. Their figures settled the astronomical unit.

The constant in the arithmetic above is where that line ends. In 2012 the IAU stopped measuring the astronomical unit. It set the unit at 149 597 870 700 m exactly. The division that made Rømer's shadow run late is the one that fixes it.

About this page

The chord comes from the law of cosines. The test file gets the same distance from Cartesian coordinates. It takes a path of its own. The same file pins 1 au = 499.004784 light-seconds. A second test file pins every delay and swing in the table, all eight rows, to the exact strings it prints. How I check things.

Sources.

• Rømer's announcement appeared in the Journal des Sçavans of 7 December 1676. That original could not be consulted here. Its English rendering is "A demonstration concerning the motion of light, communicated from Paris, in the journal des scavans, and here made English", Philosophical Transactions 12(136), 1677, pp. 893–894 — the record was checked, the text was not, so the figure of 22 minutes is reported here rather than verified.
• BIPM, The International System of Units (SI), 9th edition — c, and the astronomical unit in Table 8.
• IAU, Resolution B2 (2012), on the re-definition of the astronomical unit of length — 149 597 870 700 m exactly.
• NASA/JPL Solar System Dynamics, Approximate Positions of the Planets — semi-major axes and eccentricities, J2000, valid 1800–2050.
• J. H. Thomson, J. E. B. Ponsonby, G. N. Taylor and R. S. Roger, "A New Determination of the Solar Parallax by Means of Radar Echoes from Venus", Nature 190, 1961, pp. 519–520.
• G. H. Pettengill and others, "A radar investigation of Venus", The Astronomical Journal 67, 1962, p. 181.
• D. O. Muhleman, D. B. Holdridge and N. Block, "The astronomical unit determined by radar reflections from Venus", The Astronomical Journal 67, 1962, p. 191. These three records were checked one by one; the papers themselves could not be opened, so the observing groups and dates are reported here rather than verified.