Step Thirteen: Determine Planetary Density, Radius, and Surface Gravity

The physical size of a planet depends on not only its mass, but its physical composition. In this step, we will determine the density of each planet, which will immediately give us its radius and surface gravity.

Procedure

The density of a planet is a measure of its mass per unit volume. We will express planetary density in comparison with Earth. Thus, a planet with a density of 1.0 is exactly as dense as Earth. To convert to the usual units, multiply by 5.52 to get grams per cubic centimeter.

Leftover Oligarchs and Terrestrial Planets which form inside the snow line are made primarily out of silicate compounds (i.e., rocks). The heat of the accretion process tends to cause heavy metals, especially nickel and iron, to separate out and settle into a planetary core. The density of the fully formed planet will largely depend on the amount of nickel-iron available, and so on the size of this metallic core. To estimate the density of a Leftover Oligarch or Terrestrial Planet, roll 3d6 and apply the following:

Here, D is the planet’s density, and M is its mass in Earth-masses. Round the density off to two significant figures.

In some cases, a rocky Leftover Oligarch may have significantly higher density due to a massive impact event late in the process of planetary formation. The impact scatters most of the lower-density rocky material out into space, leaving behind a planetary body dominated by the nickel-iron core. In our own planetary system, Mercury appears to have undergone such a process. To determine whether a rocky Leftover Oligarch has unusually high density, roll 1d: on a 5 or 6 the planet will be dominated by its metallic core. Add 0.4 to the density computed above.

Failed Cores, and Leftover Oligarchs and Terrestrial Planets that form outside the snow line, incorporate a great deal of water and other ices. The density of the fully formed planet will be significantly lower than that of a rocky planet of the same mass. Determine the density of these planets by rolling 3d6:

Again, D is the planet’s density, and M is its mass in Earth-masses. Round the density off to two significant figures.

Gas Giants are constructed almost entirely out of hydrogen and helium gas. Although a gas giant will have a solid core of stone and ice, the factor dominating its density is the degree to which its gaseous envelope is compressed under gravity. To determine the density of a gas giant, let M be its mass. Then use the appropriate formula below:

Here, D is the density of the gas giant planet. Round the density off to two significant figures.

Radius

The radius of a planet (normally measured in kilometers) is dependent solely on its mass and density. If M is the planet’s mass (in Earth-masses), and D is the planet’s density, then:

Here, R is the planet’s radius in kilometers. Round off to three significant figures.

Surface Gravity

The surface gravity of a planet, measured in comparison to standard gravity at Earth’s surface, is again dependent solely on its mass and density. If M is the planet’s mass (in Earth-masses), and D is the planet’s density, then:

Here, G is the planet’s surface gravity. Round off to the nearest hundredth of a gravity. Note that under our model, a Gas Giant of 200 Earth-masses or less will always have a surface gravity of exactly 1.

Examples

Arcadia: The computations here are very straightforward. Alice adds more columns to her table and generates planetary densities, adjusting these to taste and then computing planetary radius and surface gravity for each planet.

Radius	Planet Type	Planet Mass	Density	Radius	Gravity
0.09 AU	Terrestrial Planet	0.88	0.92	6280	0.91
0.17 AU	Terrestrial Planet	1.20	1.04	6680	1.09
0.30 AU	Terrestrial Planet	0.95	1.02	6220	1.00
0.57 AU	Terrestrial Planet	1.08	1.04	6450	1.05
0.88 AU	Terrestrial Planet	0.65	0.92	5670	0.82
1.58 AU	Leftover Oligarch	0.10	0.67	3380	0.36
2.61 AU	Planetoid Belt	N/A	N/A	N/A	N/A
4.40 AU	Large Gas Giant	480	0.22	83000	2.85
5.76 AU	Medium Gas Giant	120	0.091	70000	1.00
9.50 AU	Small Gas Giant	22	0.21	30000	1.00

Beta Nine: Bob also has no difficulty with the necessary computations.

Radius	Planet Type	Planet Mass	Density	Radius	Gravity
0.27 AU	Terrestrial Planet	0.63	0.98	5500	0.85
0.45 AU	Terrestrial Planet	0.59	0.89	5550	0.78