Just like the temperature-versus-energy input plot from before, it is clear that a single straight-line relation would not fit all of the data. This is because different worlds have different compositions and different underlying physics that govern the relation between mass and radius -- there are phase changes! So instead of a single straight line, we fit a broken power-law between mass and radius. The transitional points between each pair of relations are fitted as a part of our model, which essentially means that we are performing classification of these worlds.

The first transition happens at 2 Earth masses, corresponding to about 1.23 Earth radius. We labeled the category below this point as the “Terran worlds”, because the Earth is a typical member. Many Solar System worlds, including Mercury, Venus, Mars, as well as the moons and dwarf planets, all belong to this category, exhibiting a tight relationship between mass and radius. Its neighboring category is labeled as the “Neptunian worlds”, containing three of the outer planets in the Solar System—Saturn, Uranus, and Neptune. Worlds in this category possess significant volatile envelopes, which makes their radii increase much more rapidly than the radii of the Terran worlds. Saturn sits on the verge of the Neptunian category, just to the right of the second transition, which occurs at 0.41 Jupiter mass (or around 130 Earth mass). In fact, using the classification derived in our project, Saturn is 95% likely to be a Neptunian world and 5% likely to be a Jovian world (Jupiter-like). The next category—“Jovian worlds” is kind of special. In this category, all the worlds have almost the same radius, despite the fact that mass has increased for more than 100 times. This category contains the most massive kind of planets and brown dwarfs. Some classification regards planets and brown dwarfs to be different. However, as is shown in the plot, brown dwarfs follow the same mass-radius relation as the most massive planets. The last transition is at 0.08 Solar mass. Above this limit, the internal pressures are sufficiently high that its core will be able to ignite a hydrogen nuclear fusion chain reaction. Hence the last category, “Stellar worlds”, contains stars.

Aside from classification, this mass-radius relation can also be used for forecasting. Through the past two decades, astronomers have discovered thousands of extrasolar worlds. But most of these detections measure either the mass (e.g. radial velocity) or the radius (e.g. transit). The mass-radius relation allows us to predict the mass (radius) with the other measurements.

Looking at data in this way leads to surprising implications. Brown dwarfs are merely high-mass Jupiters. Dwarf planets, like Pluto, are merely low-mass members of the same class containing the Earth. Perhaps the most surprisingly result is that the divide between Neptunian worlds and solid planets like the Earth occurs at just 2 Earth masses. This leaves very little room for Super-Earths and suggests gaseous planets occur at much lower masses than we initially expected. Effectively then, the Earth is the Super-Earth we have been looking for all along.