Rung 03 · 1609 – 1619 · Prague & Linz

The Shape of the Solar System

Johannes Kepler · a perfect map with no scale bar

A complete scale model of the solar system — every planet's relative distance precisely known — built from nothing but orbital timing, with no real units at all.

Kepler's perspective solar system: all orbital periods are measurable from Earth, and T²∝d³ gives every planet's distance as a ratio of Earth's (in AU). But the actual length of 1 AU remained unknown.

Johannes Kepler spent years wrestling with the observations of Tycho Brahe — the most precise planetary data ever recorded by naked eye — and found hidden in them a beautiful mathematical law.

He discovered that a planet's orbital period (the time for one full trip around the Sun) and its distance from the Sun are not independent: they are locked together precisely. Square the period, cube the distance, and you always get the same ratio — for every planet.

T² ∝ d³ · Period² is proportional to Distance³

Astronomers had measured precise orbital periods for every known planet just by patient watching and counting. Kepler's law converted those periods into relative distances: Mars orbits at 1.52× Earth's distance, Jupiter at 5.20×, Saturn at 9.58×. The entire solar system now had a correct shape and scale — every proportion right.

But all distances were expressed in AU (Astronomical Units) — multiples of Earth's own unknown distance from the Sun. It was like having a perfectly accurate map with no scale bar. The shape was right, but nobody knew if 1 AU was 100 million or 200 million kilometres.

The solar system had shape. It just needed one real measurement to give it size — and that would come from Rung 4. One anchor point, and Kepler's law instantly converts the whole map into real kilometres.

See that shape at its real proportions →

⚑ Assumptions that made it work

Planets orbit the Sun — the heliocentric assumption. In a geocentric model, Kepler's law simply does not appear. This was still deeply controversial in 1609. Galileo's observations supported it; the Church opposed it. The entire rung collapses without Copernicus.
Orbits are ellipses with the Sun at one focus (Kepler's First Law). Two millennia of circular-orbit tradition had required endless fudging with epicycles. Kepler accepted the slightly ugly truth that they are ellipses, and everything suddenly became clean.
The law applies universally to all planets. Kepler found this empirically and trusted it to hold everywhere. There was no physical reason yet to believe this — Newton's gravitational explanation came 66 years later and showed why it must be true.
Planetary periods can be measured precisely from a moving Earth. This requires knowing which synodic periods (planet-to-planet) to convert to sidereal (sun-centred) periods — itself requiring the heliocentric model.

✋ Try it yourself — draw an orbit, then find Kepler's law

Two pins, a loop of string and a pencil. Kepler spent years fighting the shape below; you can have it in a minute.

Push two pins into a sheet of paper on top of a cork board or a thick magazine, a few centimetres apart.
Tie a piece of string into a loop and drop it over both pins. It wants to be a fair bit longer than the gap between them.
Put your pencil inside the loop and pull it taut, then run the pencil all the way round, keeping the string tight the whole time.

You have drawn an ellipse. Now move the pins closer together and draw another; move them further apart and draw a third.

What the pins are: those two points are the foci, and they are what an ellipse is made of. Pins nearly touching gives you a circle. Pins far apart gives a long squashed one. Every orbit in the solar system is one of these.

Now the part that cost Kepler years of his life. The Sun sits at one pin. The other pin has nothing at all at it. No star, no centre, no object — just a bare point in empty space that the planet's path is nonetheless shaped around. He tried for a long time to avoid this conclusion, because it is ugly, and in the end the measurements would not let him.

Now the law itself. Kepler had no telescope for this part — he had a table of numbers and he hunted through it for a pattern for years. You can do the same hunt in five minutes, because the numbers are in this site's own catalogue. Take each planet's year in Earth-years and its distance in AU. Square the year. Cube the distance.

planet	year²	distance³
Mercury	0.058	0.058
Venus	0.378	0.378
Earth	1.000	1.000
Mars	3.538	3.538
Jupiter	140.7	140.8
Saturn	867.7	867.3
Uranus	7,059	7,066
Neptune	27,149	27,189

Check a few yourself and watch the two columns agree. Eight planets, from one that takes 88 days to one that takes 165 years, and the worst disagreement in that table is a tenth of one per cent. That is not a coincidence you can talk your way out of — it is a law, and finding it is the whole of this rung.

Then notice what you are holding. You now know exactly how much further out Neptune is than Earth, and you have not measured a single kilometre. Every distance is in "Earths". A perfect map with no scale bar — which is precisely the problem rung 4 has to solve.

The long version, if you have a few months. Everything above is indoor work. The observation the whole rung exists to explain is outdoors and takes patience: find Mars, and once a week sketch where it sits against the pattern of stars behind it. Keep going. Sooner or later it will slow, stop, and travel backwards for a few weeks before turning round again.

Nothing has gone wrong with Mars. That backwards loop is just what it looks like when the Earth, on a faster inside track, overtakes it — and explaining that loop is what forced the whole solar system to be rearranged around the Sun. The planetarium knows when the next one starts, from where you are.

When is the next backwards loop? →

The Shape of the Solar System

When Kepler Discovered His Law...