Some massless equations for world-builders

Recently I ran across some maths online which involved solving familiar equations with unfamiliar variables. Like, say I know escape-velocity but I want the surface-gravity. Or, say I know the surface-gravity (and radius) but want the density. I don't have the planetary mass; or maybe I do, but it's for a big planet and I don't want to plug those massive numbers into my calculator.

Surface gravity = GM/R²; and for escape velocity, 0.5v² = GM/R. So

surface gravity = 0.5*ve2 / R. Solving for escape velocity: root(2Rg).

Why this isn't on Infogalactic or Wikipedia yet... feh. Maybe someone should put them there.

Note that if gravity seems the same on the new surface, perhaps because you're trying to make all this Earth-familiar: mass and, more-to-the-point, density aren't directly involved. You'll be working with those factors, later. It is volume (or, radius) which drags the escape velocity higher. Fortunately, as mentioned for Venus: if I rise above the clouds I am no longer on the planet's radius, so I enjoy a lower G and a lower v_e.

Now let's do gravity, radius, and density!

Surface gravity = GM/R² again; density, M/V = M/(4π R³ / 3) = (M/R²) * 3/(4π * R).

M/R² = 4π * R * D / 3. Might as well G * 4π / 3 = 2.7956 E-10

So Surface gravity = 2.7956 E-10 * R * D. Better:

D = 3577008586 * g / R.

Density estimate for Earth at 9.807 and Don Pettit's 6670 km radius: 5255.6 kgm^-3. Not far less from the value I get on the internets 5515.3. But Earth isn't a sphere and other authors prefer 6371 km blah blah blah.

World builders are generally referred to the density of silica and various metals, but uff to all that. World builders should start instead with the actual building-blocks: the density of protoplanets in our solar-system. That's 3500 for stone meteorites and 8000 for iron. I think you can dip below 3500 if you assume lighter stone or lava. Certainly if you coat your planet with ices, water, and/or maybe water-ammonia.