Sweet Spots in Gravity Wells for the purposes of fiction

[Grarr McGrarr]

I'm a writer and I like to play in the science fiction sandbox but keep a firm grasp on what is possible. I recently started working on some notes for a setting in which we have begun to colonise the solar system.

To the question at hand. The further you fall into a gravity well, the more you feel the effects, yes? but if you start passing the material that produces the well, you become 'lighter' because you are being pulled in different directions.

Here is the specific. Trying to find the closest to Earth's gravity on the moon. My moon colonists want to burrow down into the moon's core to create large caverns for a city to be built in and to nick resources. Assuming that drilling through is no issue (thanks to mcguffin superdrill) at which point will the cavern be closest to Earthlike gravity. Also... if the cavern is pressurised (or lined and pressurised) will the mass of moon above the cavern compress the atmosphere contained in the habitation?

Please have a layman's translation in any answers... :)

 

[DEvens]

... at which point will the cavern be closest to Earthlike gravity. :)

At the surface.

Also... if the cavern is pressurised (or lined and pressurised) will the mass of moon above the cavern compress the atmosphere contained in the habitation?

Will rock still weigh something on the moon? Yes.

 

[Bandersnatch]

The further you fall into a gravity well, the more you feel the effects, yes? but if you start passing the material that produces the well, you become 'lighter' because you are being pulled in different directions.

What you wrote here (and it's correct) means that as you begin to dig down, gravity falls. So you contradict yourself in your next paragraph.
So, as noted by DEvens, you won't get more gravity than at the surface*.
The physics here basically translates to the simple fact that, as you dig down, all the matter above the radius you've dug to can be treated as non-existent for the purposes of calculating gravity. It's just as if you were standing on the surface of a smaller planet.

*Actually, you could get a bit more, depending on the exact composition of the moon. The simple reduction of gravity with radius works for uniform sphere of material, and the Moon is not perfectly uniform. However, the increase won't be much, if at all, so it's not worth bothering.

Also... if the cavern is pressurised (or lined and pressurised) will the mass of moon above the cavern compress the atmosphere contained in the habitation?

No, without a continuous column of atmosphere like on Earth, the rocks above won't matter. But, you can probably rather easily pressurise the cavern to your needs - after all, it has to be airtight anyway.

 

[Khashishi]

If the density of the moon were completely uniform, then the gravity decreases linearly as you go below the surface, reaching 0 at the core. But really, the density is probably somewhat greater in the center of the moon. This will cause the gravity to decrease more slowly at first (as you descend from the surface), but then catch up later on, still reaching 0 at the core. With a very nonuniform (unrealistic) density, the gravity could even increase as you descend over some interval.

The pressure is basically just the weight of the layer of rocks above your point per unit area. If you get too deep, then the pressure will create engineering hurdles for your cavern liner, which has to support the difference between the interior and exterior pressure.

 

[Grarr McGrarr]

Thanks. That leads to two follow ups... one I think is quick, the other not so much.

1) so gravity is (roughly) strongest at the top of the tallest mountain? Or on Earth, at the top of the atmosphere?

2) I remember reading that the smallest amount of matter that would compress into a black hole was roughly the size of a small city. Any less and there would not be enough mass to form the effect of trapping light etc (this obviously requiring you to be able to compress a city down to that size in the first place). With this in mind would it be possible (given the magic crushin mcguffin) to take the matter carved from the cavern and compress it into a super dense material so that would have a greater gravitational pull on those standing on it.
If so, would carving such huge lumps out of the moon and turning them into floor tiles destabilise the moon or the gravity well?

I toyed with the idea of just crushing the moon down until it is dense enough to have 1 G gravity. I thought that would be problematic for my setting for a few reasons but none scientific (other than the need for said big magic crusher) How big would that be? How would a smaller, denser moon effect the Earth?

 

[Khashishi]

1) so gravity is (roughly) strongest at the top of the tallest mountain? Or on Earth, at the top of the atmosphere?

No, of course not. We were talking about a smooth ball before. Gravity is weaker below the surface because mass above you pulls you upward and cancels with some mass below you. This argument does not apply to a mountain.

2) I remember reading that the smallest amount of matter that would compress into a black hole was roughly the size of a small city.

Wrong. There is no known limit on the size of a black hole. The LHC was predicted by some to create micro black holes out of single particles. Small black holes evaporate very quickly. But a city is very large by quantum scales.

But, yes, if you had a magic crushin mcguffin, you could create 1G gravity in the moon by creating dense spots. It would require technology far beyond what we currently possess. The gravitational field of the moon far from the moon wouldn't be affected. It wouldn't affect the orbit of the moon around the Earth or affect the Earth in any significant way.

 

[Grarr McGrarr]

This argument does not apply to a mountain.

I wasn't referring to the mountain as a singular object but as part of the whole Earth. Standing at the base of the mountain obviously the mass of it is to yourside and above and it's pull would drag you in that direction where as if it is below your feet then the drag is in (roughly speaking) one direction. Then I thought, whilst we are talking about minutia, the Earth surface could be considered the top of the gases rather than the first rock surface so the point where there is maximum mass below you would be at the edge of our atmosphere.

Small black holes evaporate very quickly. But a city is very large by quantum scales.

Yes, I believe this was the point behind picking a city. That the evaporation of the black hole (is it me, or is that a VERY misleading term?) isn't instant but leaves a persistent object..
or
We are talking about simple mass compared to what happens in the LHC where massive amounts of energy are added to the lightest particles.

Honestly, I don't know... I'm recalling an example from a while ago given by someone who (I think) knew what they were talking about to me, someone doing their best to grab hold of the concept. Regardless, that was an example not the core point.

As to the 'crunched moon' any idea how big it would be if the moon was reduced to a size of 1G? I'm now thinking of the inside of the moon being modified to look like an orange on the inside, the segments being crushed down to create a 1 G surface whilst the surface remaining the same (for the purpose of aesthetics and advertising hoarding space :P)

 

[Khashishi]

The radius of the moon would be $R_{Earth} \sqrt{\frac{M_{moon}}{M_{Earth}}}$ or about 707 km to have surface gravity of 1 gee.

For the purposes of planet gravity, you should just ignore mountains and treat the Earth as perfectly smooth. This is a very good approximation. Things are more complicated when you start considering strange shapes, and it's not always true that gravity increases or decreases going up, but you have to go through the calculus and check.

 

[Czcibor]

If you:
-like deep drilling
-want to colonize nearby bodies
-need some good justification for that drillingI'd suggest you trying your luck on Mars. If you dig there, you'd get higher atmospheric pressure at no effort, which may allow your colonizers to change space suits for merely breathing apparatus.

 

[Grarr McGrarr]

The radius of the moon would be $R_{Earth} \sqrt{\frac{M_{moon}}{M_{Earth}}}$ or about 707 km to have surface gravity of 1 gee.

Thanks muchly. :)

 

[Grarr McGrarr]

If you:
-like deep drilling
-want to colonize nearby bodies
-need some good justification for that drillingI'd suggest you trying your luck on Mars. If you dig there, you'd get higher atmospheric pressure at no effort, which may allow your colonizers to change space suits for merely breathing apparatus.

The setting involves stations / settlements at all the lagrange points, orbitals around Earth, the Moon, Venus (with cloud cities) and Mars (with surface and subsurface colonies) and numerous bases and outposts in the asteroid belt. However, trips to other planets are seasonal as they aren't always close to Earth. The Earth / Moon system therefore gets the most love. I'm currently drawing parrallels with the early expansion of european powers into the carribean in the 1640 - 1660 period. The old world (literally) sending out expeditions for years at a time to plunder the natural resources and to lock down strategic locations.

The moon in this instance is the edifice of the big bad... all the tech, all the power and just an intimidating presence. Something for the players to fear. Mars is a place for them to run away to. :)

 

[newjerseyrunner]

Gravity is very weak. When Charlie Mason and Jeremiah Dixon were surveying Pennsylvania, they use the most advanced surveying instruments at the time, the instruments used gravity to find straight down and allowed for very accurate measure of distance and angles. When they surveyed, they went one way, then back, in order to compensate for any bias to either direction from the instrument. They kept running into a problem, their instrument seemed to have a bias north, not left or right, but north, regardless of if the instrument was pointed east of west. It took over a year for someone from the Royal Society to realize that the Allegheny Mountains's gravity was pulling on the instrument.