The previous post was about Kepler’s observation that the planets were spaced out around the sun the same way that nested regular solids would be. Kepler only knew of six planets, which was very convenient because there are only five regular solids. In fact, Kepler thought there could only be six planets *because* there are only five regular solids.

The distances to each of the planets is roughly a geometric series. Ratios of consecutive distances are between 1.3 and 3.4. That means the distances should be fairly evenly spaced on a logarithmic scale. The plot below shows that this is the case.

The plot goes beyond the six planets known in Kepler’s day and adds four more: Uranus, Neptune, Pluto, and Eris. Here I’m counting the two largest Kuiper belt objects as planets. Distances are measured in astronomical units, i.e. Earth = 1.

**Update**: The fit is even better if we include Ceres, the largest object in the asteroid belt. It’s now called a dwarf planet, but it was considered a planet for the first fifty years after it was discovered.

## Extrasolar planets

Is this just true of our solar system, or is it true of other planetary systems as well? At the time of writing, we know of one planetary system outside our own that has 8 planets. Three other systems have around 7 planets; I say “around” because it depends on whether you include unconfirmed planets. The spacing between planets in these other systems is also fairly even on a logarithmic scale. Data on planet distances is taken from each system’s Wikipedia page. Distances are semimajor axes, not average distances.

### Kepler-90

Kepler-90 is the only planetary system outside our own with eight confirmed planets that we know of.

### HD 10180

HD 10180 has seven confirmed planets and two unconfirmed planets. The unconfirmed planets are included below, the 3rd and 6th objects.

### HR 8832

The next largest system is HR 8832 with five confirmed planets and two unconfirmed, numbers 5 and 6 below. It would work out well for this post if the 6th object were found to be a little closer to its star.

### TRAPPIST-1

TRAPPIST-1 is interesting because the planets are very close to their star, ranging from 0.01 to 0.06 AU. Also, this system follows an even logarithmic spacing more than the others.

#### Systems with six planets

There are currently four known systems with four planets: Kepler-11, Kepler-20, HD 40307, and HD 34445. They also appear to have planets evenly spaced on a log scale.

Like TRAPPIST-1, Kepler-20 has planets closer in and more evenly spaced (on a log scale).

Some of these graphs indeed look pretty straight.

But even a uniform distribution looks roughly straight: https://www.strandmark.net/Planets.html So some test might be appropriate.

This looks close to random to me – here’s a simulation of 10 planets, with distances drawn uniformly from [0…1] for each planet.

Raw distances

https://imgur.com/a/by2Kd

Normalized by distance to furthest planet.

https://imgur.com/a/1GOe4

It all really makes sense for this to happen. From the moment a star forms its gravitational force is pulling on the gas and dust that form planets i believe if a star forms planetary bodies it is because of how the gravitational forces pull and tug on them and if it was just a random formation there would be major collisions and planets would never last i also find that from in-word to outer planets you will see small to gas giants and back to smaller ice planets each planet is held in place by gravity i wonder if the bigger planets help with a tug of war and keep them all within a certain distant from its parent star

After writing this post, a few people told me this is the Titius-Bode law. Titius and Bode observed a couple centuries ago that the planets in our solar system are roughly geometrically spaced. This was dismissed as coincidence, but evidence from other solar systems implies that there’s something to it. Of course it’s not a strict law, but it does describe a common pattern.

It seems to have something to do with gravitational resonances. Some configurations are unstable and so we’re unlikely to see those.

Absolutely killer post.

I wonder if there’s also a correlation between mass. The intuition is that if two big planets are close together, they eventually collide. So a scatterplot with the planet order and mass vs. distance from the sun might show something interesting, if you can fit it to a surface.

“Spaced” planets. I get it!

This makes perfect sense to me. Gravity shapes spacetime. The closer a planet is to the sun, the more gravitational energy there is, the faster the planet moves. The planet will carve out a groove in space proportional to its mass and matter will fall into the groove caught by the gravity well. Because of the speed of the planet the grooves will be smaller closer to the sun. The locations of planets are at the bottom of the grooves – the lowest energy state balanced between other energy states, a result of the geometry of space between the gravity wells. This should be easy to simulate – I’d be surprised if someone hasn’t already done it.

It has been simulated, and approximate logarithmic spacing often shows up in the simulations. On the other hand, you don’t always see logarithmic spacing, and when you do see it, it isn’t exact. So there’s more to explain.

Just from the plots I made, it seems that logarithmic spacing holds more precisely for systems with planets close to their star, like TRAPPIST-1 and Kepler-20.