Loading the Canon: Surface Gravity

Warning: Lots of number crunching follows.

In the last post I talked about orbits and habitable zones as shown in the Map and the 'Verse in Numbers. Another issue that comes up consistently though ViN is the stated planetary masses and the assumed surface gravity. Every time we see Our Heroes on the ground though the series and BDM they're on a world with Earth Normal gravity. At least close enough to it that they're walking and moving as if everything's great.

That's OK. Trek and Star Wars did that too. It saves on the FX budget and doesn't hurt the story telling. What it does do, though, is raise the question of why you have Earth normal surface gravity on a Ceres-sized moon.

Calculating the surface gravity of a world is another straightforward formula. There's a lot of places to find it, but you can see the formula and a chart showing the surface gravity for worlds in the Sol system here.

From that site, we have the following formula:

The surface gravity (g) of a body depends on the mass (M) and the radius (r) of the given body. The formula which relates these quantities is:

g = G * M / r²

where G is called the Gravitational constant.

When calculating the surface gravity using this formula it is best to stick to the MKS system where the units for distance are meters, the units for mass are kilograms, and the units for time are seconds. In this system, the gravitational constant has the value:

G = 6.67 x 10^-11 Newton-meter²/kilogram².

We're going to use SI measurements (That's International System of Units, not Sports Illustrated) for our calculations. Running the numbers for Earth first for a baseline, we get:

(6.67*10^-11) * (5.98*10^24) / (6.378*10^6)^2 = 9.805234578

Where:
(6.67*10^-11) is the gravitational constant.
(5.974*10^24) is the Mass of Earth in Kilograms
(6.378*10^6)^2 is the square of Earth's radius in Meters.
and 9.8 is the surface acceleration in Meters per Second.

Now, hunting around I've found various numbers to plug into the calculation which gives a slight range of values. For our purposes, we'll go with these numbers because they're close enough for our needs.

The 'ViN gives us a diameter and mass for each of the worlds of the 'Verse. That gives us numbers to plug into each value. Since Seana is from Ariel, I'll use Ariel and one of its moons to crunch some numbers here.

According to ViN, Ariel has a diameter of 13,000 km and a mass of 6.323 * 10^21 metric tons, or 6.323 * 10^24 kg. Plugging this into the formula, we get:

(6.67*10^-11) * (6.323*10^24) / ((13.0/2)*10^6)^2 = 9.982108876

That's just a touch over Earth normal gravity, which is good. Ariel is about the same mass and diameter as Earth, so we'd expect to get roughly the same surface gravity.

So now let's try a moon. Poseidon is Ariel's third moon and is given a diameter of 1024 km and a mass of 3.563 * 10^22 kg. Plugging this in we get:

(6.67*10^-11) * (3.889*10^22) / ((1.024/2)*10^6)^2 = 9.895183563

A bit less than Earth normal, so again we're good. But are we?

Earth has an average density of 5.51 g/cm³ which is about double that of granite and over 5 times that of water. Much of that density is the result of Earth's molten iron core, which has a density of around 8 g/cm³. The ViN postulates that they increased the surface gravity of the smaller worlds by using gravity tech to somehow compress them. Same mass with a smaller diameter gives you a higher surface gravity. But it also increases the density of the target. How much? Let's see.

Density is Mass / Volume. The formula to calculate volume of a sphere is 4/3*π*radius³ which we'll want to work in cm, rather than Kilometers. So Earth's 6.378*10^3 km becomes 6.378*10^8 cm. Plugging that into our formula, we get 1.086781293e+27 cm³. Our mass of 5.974*10^24 kg becomes 5.974e+27 g. This gives us:

(5.9736e27) / (4/3 * 3.14159 * (6.378e8)^3) = 5.496967 g/cm³

This is comfortably close to the 5.51 g/cm³ stated above. The difference between using Earth's mean radius versus its equatorial radius.

Ariel itself being an Earth-like planet in mass and radius will have a very similar average density. But what about the "compressed" Poseidon? Let's crunch the numbers.

Mass: 3.889e22 kg (e25 for g)
Radius: 1024 km / 2 is 5.12e2 km (e7 for cm)

(3.889e25) / (4/3 * 3.14159 * (5.12e7)^3) = 6.91735107e1 g/cm³ or around 69 g/cm³

That's a LOT higher than Earth's density. Even solid Osmium, the highest density "normal" material, only has a density of 22.57 g/cm³. Now, 'compressing' a moon down to give it an Earth Normal surface gravity is probably more rational than installing artificial gravity generators to get the same effect. Not that either of these solutions is especially rational.

I could run more examples, but then, you can run them yourself and see what I mean.

Conclusions:

What does this mean to us as gamers in the 'Verse? Well, for starters, if you're just compressing the existing body then putting a mining colony on a small moon is a patently stupid idea. The denser the rock, the harder it is to mine. That's just a simple fact. Our little moons have a density three times higher than Osmium. Erm. . . . no.

While Canon gives us Earth Normal surface gravity on all the worlds of the 'Verse, they never explained it in the show or movie. ViN gives us 'compressed' moons that have a high enough density to provide the gravity we want. But, as we've seen, gives us an insanely high average density. The alternative is artificial gravity, which the campaign has but it was never implied that it could be used on a planetary scale.

Now, we could work with a compressed core with a thinner layer of normal density crust above it. That lets us mine our small moons to a depth of kilometers to tens of kilometers, and possibly even arrange it so the more valuable minerals 'float to the top' during the compression process, enriching the crust.

While this doesn't appear to be presented anywhere in the published materials, it is a solution we can work with and one I'm going to apply to Hale's Moon from a story perspective. Not that it'll ever actually come up.

So, to sum up, worlds smaller than Earth wind up with an Earth Normal gravity by compressing the entire body to increase its average density. The compression is somehow stratified, so the outer layers maintain a more or less normal density and only the inner layers are crunched down like Unobtanium.

Scientifically sound? No. I'd have to run numbers I don't remember, but gut instinct tells me materials with a density of 69 g/cm³ wouldn't actually be stable under normal conditions. But within the confines of an RP environment? Sure. Why not?