A lot. The strength of Earth’s gravity is controlled by two fundamental properties of our planet; the distance from the very core of the earth to the surface, and how much mass is held within that space. If our planet were the same size, but made out of packing peanuts instead of rock, the force of gravity at the surface would be much less than it currently is. Similarly, if we took the same amount of material – the same mass – but changed how densely packed it is, you could reduce the pull of gravity on the surface. However, neither of these things is easily changed. The Earth is Earth-sized because it’s mostly made of molten rock and metal, with a bit of liquid water and solid rock on the surface. Molten rock and metal are both pretty hard to compress beyond a certain density, and difficult to fluff up to make it more styrofoam-like (unless you fill it with pockets of gas).
The equation to figure out the strength of gravity on the Earth is pretty simple: g = GM/r^2. M is the mass of the planet, r is the distance from the center to the surface, and G is the gravitational constant, which is a constant feature of our Universe. It’s also a very small number, so it winds up canceling out the very large numbers the Earth is going to dump into this equation. Once we plug in the radius of the Earth and the mass of the earth, we find that gravity on the surface of the Earth is pulling you towards the ground at 9.81 m/s every second.
If we wanted to change the force of gravity, we’d have to reduce this number, which means either increasing the size of the planet (basically impossible), or removing some of its mass (more possible). We have our method of mass removal given in the question, so we’re going to build rocket ships, load them up with stuff, and launch them out into space. How much stuff would we need to remove before the Earth’s gravity changes? Technically, everything we send up removes some mass from the Earth, but it’s such a minuscule fraction of the Earth’s mass that we will never notice the difference. So how much material would we have to send up before we’d notice the difference?
Let’s try and change gravity by ten percent.
This means everyone will feel 10% lighter on the surface, and with the same amount of force, you’d be able to jump higher, and falling would hurt less.
This means we need to reduce the Earth’s gravitational pull from 9.81 meters per second per second to 8.83. If we’re not expanding the distance to the Earth’s surface, the only thing left to change is the mass of the earth, so we’ll have to reduce the Earth’s mass by ten percent. Pretty straightforward.
But the Earth is pretty big. 5.972 × 10^24 kg big. This is a number so outrageously huge that it basically doesn’t make sense to write it in kilograms anymore. We typically write it in “Earth masses” instead, but that’s even less useful for getting a sense of scale. In any case, let’s divide this number to find 10% – there’s some useful scale coming ahead. 10% of the Earth is 5.972 × 10^23 big – one less zero, but twenty three zeros is still a pretty big number.
In fact, it is the mass of Mars. Mars is only slightly more massive than this – with a mass of 6.39 × 10^23 kg, it’s just under 11 percent of the mass of the Earth. So in order to change the gravity of the Earth by a noticeable but not incredibly dramatic ten percent, we would have to extract from the surface of the earth, One Whole Mars worth of material. This, as you can probably guess, would be grossly unwise. If we were to peel off the entire crust of the earth, which is some 3-30 miles deep, and throw in the entire mass of all of the oceans for kicks, we’re still only looking at about a half a percent of the earth’s mass, and we’ve made our planet into Lava Planet. (Never mind the mechanics of peeling off the crust of the Earth, which I can only imagine would go spectacularly poorly.) In fact, in order to get our ten percent, we’d have to extract pretty much the entire upper mantle and jet it into space in order to reduce gravity by 10 percent, and our surface relies on that upper mantle for stability. When the mantle moves, our crust moves with it- which is part of the reason we get earthquakes. Removing that structure from underneath us would be a Grade A Bad Plan.
Of course, beyond the matter of extracting a Mars-worth of magma from the innards of the planet, there’s the slight issue of where to put it. Mars is not exactly our smallest planetary body, so if we reassembled all of our Earth-extractions into a planet again we might run into some minor orbital disturbances, suddenly having a second Mars hanging around. If we don’t leave it as a single object, but leave it scattered in small pieces, then we have created our very own Asteroid belt. I would recommend putting your asteroid belt very far away from Earth, or holy space junk Batman, we have created a very hazardous near-Earth environment, which already needs some cleaning.
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