Doesn't the inverse square law only apply to the diminishing effects of gravity over range, with the radius in question being said range?

Volume and mass are almost directly linearly correlated (granted, given the same density). With mass being directly tied to gravitational pull on a linear basis...So I think in this case you might just be using the wrong maths. Surface area of the planet would be a lot better value to terraform by. Feel free to correct me if I'm wrong, though.

I'm not using the wrong math, but we're into some moderately esoteric physics, so I'm going to have to show it to explain. If I'm misunderstanding you, then I'm preemptively sorry.

The shell theorem states that a spherically symmetric body will behave as if it is a point mass concentrated at its geometrical center, and that it doesn't matter if it is solid or hollow. So we can calculate the gravitational force on someone standing on the surface of the Earth using the mass and radius of the Earth, just as we do to work out the pull on the moon. It also states that anything inside a spherically symmetrical shell will not be affected by that shell's gravity at all. The parts closer to you are counterbalanced by the parts farther away. This means that someone digging down into a spherical planet will see exactly what someone would see if they were peeling away the planet instead.

(The second part isn't particularly relevant, but it is interesting, so I'm not going to take it out.)

Now for the math:

1. The planet's mass is equal to rho*4/3*pi*r^3, and we'll assume that rho (density) is constant. The acceleration due to gravity on a planet's surface gPlanet (assuming that the second object is small relative to the first one) is G*mplanet/r^2. If we substitute in the equation for the planet's mass we get G*rho*4/3*pi*r^3/r^2, which simplifies to G*rho*4/3*pi*r.

2. The pressure on a planet's surface P is (slightly simplified) equal to (MassAtm/4*pi*r^2)*gPlanet. If this doesn't make sense, consider conservation of momentum. The pressure on the piece of air just above the ground must be equal to the force of gravity on the column of air above that piece. Otherwise, the air would be accelerating upwards or downwards. Either would be bad if sustained.

3. Let's assume that each terraforming platform produces so many tons of atmosphere each year, dMassAtm. Using this we can work out:

dP = (dMassAtm*gPlanet)/(4*pi*r^2) = (dMassAtm*G*rho*4/3*pi*r)/(4*pi*r^2).

Cancelling terms, we get:

dP = (dMassAtm*G*rho)/(3*r)

In other words, the change in pressure is proportional to density (smaller planet for a given mass means less surface area to spread the weight out across and higher gravity) and inversely proportional to radius (bigger planet means that you have to pump out more mass). In aurora terms, we'd spec the terraforming machines on Earth, and then multiply the rate by the relative density and by the inverse of the relative radius.

Make sense?

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