User blog:Sirwhiteout/Realmspace Astronomy - Gravity of Planets by Composition

This is part 1 of 2-post series that is the result of a very interesting discussion with Lhynard and Moviesign on a realistic way to calculate the gravity of celestial bodies based on their size and elemental composition.

In Spelljammer: AD&D Adventures in Space, it is stated that all celestial bodies have a gravity of 1 "Earth" gravity, regardless of size and composition. What if one wants to do better? Say, only spelljammers have a gravity of 1 g as an effect of the magic of their Spelljamming helms, but planets and other celestial objects are subject to the laws of Physics. How to proceed?

The following is a very basic estimate of how this calculation could be carried out, with the results organized on a table for easy reference to the realist DM.

The gravitational field $g$ of a mass $m$  at a distance $R$  from the center of mass is given by

$$g = \frac{G m}{R^2}$$

where $G$ is Newton's gravitational constant. So the total mass of a planet will depend on its size and the average density of the material it is composed of.

So the first ingredient is the density of each of the four elements. Considering that fire is just very hot air, and estimating the density of earth as the average density of the planet Earth, we arrive at the following values:


 * $\rho_{\text{Fire}} = $ 0,3 kg/m³
 * $\rho_{\text{Air}} = $ 1,225 kg/m³
 * $\rho_{\text{Water}} = $ 1000 kg/m³
 * $\rho_{\text{Earth}} = $ 5510 kg/m³

Next is the planet's size. Since the volume of a sphere as a function of its radius is given by the well-known $V = \frac{4}{3}\pi R^3$, we can combine this with the gravitational field equation above to find

$$g = \frac{4}{3} G \pi \rho R$$

So now we can construct a table of a planet's gravity as a function of its size class.

The radii of all these classes were taken using the following convention: for class E the radius of Earth was used; for class D the radius of Selûne was used; the upper bound of class A and the lower body of class J were considered the radii of those classes; for all other classes their average radius was considered.

Note that an E-sized earth planet with an average radius equal to Earth's has exactly 1 g, so this is the calibration of the entire table. Also note that this is a very simplified table, assuming that air bodies have an average density equal to 1 atmosphere throughout their interiors, which is not easily justifiable physically due to hydrostatic equilibrium. Similarly, water bodies would have crushing pressures at their depths, to the point where water would no longer be liquid, instead forming an ocean floor made up of exotic forms of ice.

More on this on the next post. We will be taking a detailed look at the gravity and the atmosphere of Coliar.