Bulk Properties of the Planets II

When a planet transits its star (see here), and we’re able to determine it’s mass through some other means, typically Doppler spectroscopy (see here and here), then we can derive its density rather easily.

$\displaystyle \rho_{pl} = \frac{m_{pl}}{V_{pl}} = \frac{3m_{pl}}{4 \pi r_{pl}^3}$

Where $V$ is the volume of the planet, equivalent to $4/3 * \pi r^3$.

Planets of similar densities are not necessarily the same in bulk composition. With the bulk composition being held constant, the radius will increase with mass in a non-linear way such that the increase in the radius slows, while the density increases. After enough mass is built up, the trend reverses, with the gravity compressing the planet more as additional mass is added.

For planets made purely of one substance (half-seriously referred to as “mathematician’s planets”), one can plot their mass and radius on a diagram and see that we actually can use a planet’s observed mass and radius to infer something about its composition, assuming iron, rock, water and gas are the dominant things out of which a planet will be comprised. This seems to be a reasonable assumption based on what we know of how planets form and the observed composition of planet-forming disks.

Plotting the Solar System planets (minus Mercury and Venus) on such a diagram, we see that the radius of Earth and Mars are consistent with being almost entirely rock, with Earth being obviously too dense to be entirely made of rock, requiring an iron component. Uranus and Neptune are slightly too large for their mass to be explained by a composition of 100% water. So a (small, by fraction of mass) gaseous component must exist. Jupiter and Saturn are far too large for their mass to be explained by being mostly water in composition, so it’s clear that they are mostly made of gas.

These things are, for the most part, able to be determined through direct study of the planet themselves, but this is not easy to do for extrasolar planets. For the transiting planets, we’re afforded a radius and a mass and left to interpolate a composition based on it. A complication arises in that there are not unique solutions for the composition of a planet for some values of the radius and mass. A 100% gas planet will stick out as being purely gas, while a 100% iron planet will clearly be an enormous cannonball. But planets can have layers of all three and this can create degeneracies in models of their interior. An small ball of iron with a large layer of water and gas will have a similar radius to a large rocky planet with a small layer of gas. As a real example (and as of the time of this writing), it is not clear if GJ 1214 b is a sort of “mini-Neptune,” with a larger core and a lot of water, or a sort of “micro Jovian,” with a small core and a large gaseous envelope. Even Uranus and Neptune show that there’s some ambiguity here. The two are believed to have solid cores under water mantles, but the mass of the rock and/or iron is hard to disentangle from the mass of the water mantle.

So it is clear that different compositions can be thrown together to blur the classification of planets into neat categories of terrestrial planets, ice giants, and gas giants. Indeed, they do not separate out very well in a mass-radius diagram.

Bulk Properties of the Planets

Even without going outside our solar system, we can take a look at the eight that are well known to even our children and get a preliminary grasp on the types of the planets that we can expect to find throughout the Universe. Looking at the graphic above, which has the size of our planetary system’s members to scale (but not their separations from each other), we readily note that the planets available to use for close study assume a wide variety of appearances and sizes. Furthermore, they visually seem to fall into two or three different classifications. Let’s take a more quantitative approach to comparing these planets.

The planets that we are most intuitively familiar with are the smallest members of the planetary zoo. They are the terrestrial planets, and are composed of rock and metal, silicates and iron. They have high densities and sudden phase transition boundaries between states of matter separating their solid and gaseous components (which we might call a “surface”). For our solar system, this corresponds to the planets Mercury, Venus, Earth and Mars.

Because Earth is the most massive representative of this type of planet that we have available to us for study, terrestrial planets of ~Earth-mass are generally called “Earth-like,” but this similarity is in bulk composition alone, and is not to imply habitability. The term is loose enough that even the infernally hellish Venus would fit the description. It’s important to remember that the differences between Earth and Venus are “skin-deep” so to speak. That which makes the two planets so different is confined to a thin layer of gases clinging onto their surfaces, and is a negligible fraction of the planet’s total mass. Planets that are more massive than Earth, typically about twice the mass of Earth and above, are called “super-Earths,” but the term is a loose one.

Being the prototypical terrestrial planet, the mass of Earth defines one of the more commonly used mass measurements. One “earth-mass,” $M_\oplus$, is a mass unit of mass equivalent to the mass of Earth.

Moving further out in the Solar System, we find the gas giant planets Jupiter and Saturn.

The Bulk Composition of the Gas Giants by Mass

These planets are those who have the majority of their mass in the form of “gases,” or at least those elements labelled so on the Periodic Table. As the depth into the planet increases, the atmospheric pressure increases considerably. Deep within the planet, the pressure is great enough that the gas slowly transitions into a liquid, with exotic properties that generate enormously powerful magnetospheres. At their centre may exist a core of solid material of several Earth-masses.

As a planet-forming disk evolves, it’s composition will change such that the gas-to-dust ratio drops until all of the gas has either fallen into the star, been accreted by planets, or pushed out of the planetary system through the radiation pressure of sunlight from the star. Therefore, planets that form first will more closely resemble the composition of the disk where they were formed. We can tell therefore that Jupiter formed first as its composition most closely resembles not only the sun, but that of the protoplanetary disk out of which it formed as well. In the case of gas giants, large solid cores of several Earth-mass grow in size until they begin runaway accretion of hydrogen from the disk (this typically occurs at a mass of ~10$M_\oplus$). Furthermore, Saturn formed later, as is evidenced by the lower gas-to-rock fraction.

Jupiter is the most massive example of this class of planets in our Solar system and therefore sets another unit of mass, the “Jupiter-mass,”$M_J$, roughly equal to 318$M_\oplus$,

Further out in the Solar system lie the planets that formed later, and had access to only the scraps of what was left of the hydrogen and helium in the disk. Their compositions are dominated by the ices that condensed out of the volatiles in the disk. These planets are called the “ice giants,” and are represented by Uranus and Neptune.

Bulk Composition of the Ice Giants by Mass

Like the gas giants, their atmospheres get thicker with depth but they reach a boundary that is more compositional than arising from a state of matter, where the composition goes from an H-He envelope to an icy mantle, likely dominated by water ice with other volatiles (ammonia, methane, etc). There is likely not a quick phase transition between corresponding to a surface on such planets.

Neptune is the most massive of these planets in our Solar System, and is thus the prototypical “Neptune-like planet.” A unit of mass based off Neptune can obviously be conceived, but it is not used much in practice. With Neptune having a mass of ~17$M_\oplus$, the masses for these planets are usually expressed in terms of$M_\oplus$or$M_J$.

When the masses and radii of the Solar System planets are plotted, they group fairly nicely sorted by their type.

Mass-Radius Diagram for the Solar System

In an upcoming post, we’ll look at doing this for extrasolar planets to see how our solar system fits into the range of extrasolar planet masses and radii and how well these neat cookie-cutter categories hold up in the wealth of examples of extrasolar planets with measured masses and radii to date.