# How deep would you have to dig into Mars to reach Earth's gravity?

by Zaku2
Tags: deep, earth, gravity, mars, reach
 P: 3 I'm a huge space colonization and enterprise enthusiast. one thing i have noticed about most of the mars habitat plans is they all plan to live on the surface, which is at 38% of earth's gravity. this would lead to similar if not the same medical problems with astronauts that we already face in zero G. but in space we can create artifical gravity to compensate. I want to know how far down we would have to go to build a colony under the surface at earths normal gravity?
 P: 493 Hello, weclome to PF. There's no such depth. The gravitational acceleration of a uniformly dense spherical body falls linearly as you dig down. It'll never be higher than at the surface.(cf.shell theorem) Planets are obviously not uniformly dense, so that changes things a bit, but you're still stuck with falling gravity, even if not at one to one gravity:distance relationship.
P: 81
 Quote by Bandersnatch There's no such depth.
That's true.

 The gravitational acceleration of a uniformly dense spherical body falls linearly as you dig down. It'll never be higher than at the surface.(cf.shell theorem) Planets are obviously not uniformly dense, so that changes things a bit, but you're still stuck with falling gravity, even if not at one to one gravity:distance relationship.
No, in the case of Earth gravity is the strongest at the inner mantle/outer core boundary. I don't know the specifics about Mars but the effect is much to weak to get 1 G gravity on Mars.

PF Gold
P: 1,368

## How deep would you have to dig into Mars to reach Earth's gravity?

 Quote by glappkaeft That's true. No, in the case of Earth gravity is the strongest at the inner mantle/outer core boundary. I don't know the specifics about Mars but the effect is much to weak to get 1 G gravity on Mars.
http://en.wikipedia.org/wiki/Shell_theorem

I hope your not saying the shell theorem is incorrect ?

keep in mind the shell theorem is not describing center of mass
edit or rather center of gravity
Math
Emeritus
Thanks
PF Gold
P: 38,705
 Quote by glappkaeft No, in the case of Earth gravity is the strongest at the inner mantle/outer core boundary. I don't know the specifics about Mars but the effect is much to weak to get 1 G gravity on Mars.
No, the rate of change of gravity will be largest at the boundary between the two densities but the strength of the gravitational force will decrease as you go down.
 P: 3 Fair enough. that assumes your passing through the mass instead of removing it? what if that 'uniform' mass was not above you to counter-act the change in your distance from the shell? like a 'frustam' shaped hole excavated to the apropriate depth, and wider at the top then the bottom.
P: 493
@Mordred, HallsofIvy
This is what glappkaeft is refering to:
http://en.wikipedia.org/wiki/File:EarthGravityPREM.jpg

 Quote by Zaku2 Fair enough. that assumes your passing through the mass instead of removing it? what if that 'uniform' mass was not above you to counter-act the change in your distance from the shell? like a 'frustam' shaped hole excavated to the apropriate depth, and wider at the top then the bottom.
Without doing any calculations, I'd say you'd need to dig down larger part of the surface-centre distance, while removing large part of the planet, and somehow preventing it from rebounding to the hydrostatic equilibrium shape. Unfeasible doesn't begin to describe it.
 PF Gold P: 1,368 Ah that makes more sense. I wasn't sure if he was referring to the apparent or effective gravity or not.
P: 118
 Fair enough. that assumes your passing through the mass instead of removing it? what if that 'uniform' mass was not above you to counter-act the change in your distance from the shell? like a 'frustam' shaped hole excavated to the apropriate depth, and wider at the top then the bottom.
Assuming the hole you dig is small compared to the planet, Gauss's law tells us that we don't have to worry about the mass above you. It doesn't matter if you pass through it, or remove it.
Mentor
P: 14,242
 Quote by Mordred I hope your not saying the shell theorem is incorrect ?
Putting words in glappkaeft's mouth, or fingers, he's saying that the shell theorem does not say that gravitational force decreases with increasing depth. He's correct. The shell theorem does not say that. You have to add the assumption of a uniform density to reach that conclusion. He's also correct in that gravitational force inside the Earth reaches a maximum value at the core/mantle boundary. The gravitational force halfway down to the center of the Earth is about 9% higher than the surface value. This is because the Earth's core comprises a bit less than 1/3 of the Earth's total mass but occupies a bit more than 1/6 it's total volume.

What the shell theorem does say is that for an object with a spherical mass distribution (density is a function of radial distance from the center), it's only the mass below that counts. You can use the shell theorem to find the condition that make gravitational force increase or decrease with increasing depth. Defining ##\rho(r)## as the density at some distance ##r## from the center and ##\bar{\rho}(r)## as the average density of all the stuff below that distance, you should find that gravitational force increases with depth if ##\rho(r) < \frac 2 3 \bar{\rho}(r)##.
PF Gold
P: 1,368
 Quote by D H Putting words in glappkaeft's mouth, or fingers, he's saying that the shell theorem does not say that gravitational force decreases with increasing depth. He's correct. The shell theorem does not say that. You have to add the assumption of a uniform density to reach that conclusion. He's also correct in that gravitational force inside the Earth reaches a maximum value at the core/mantle boundary. The gravitational force halfway down to the center of the Earth is about 9% higher than the surface value. This is because the Earth's core comprises a bit less than 1/3 of the Earth's total mass but occupies a bit more than 1/6 it's total volume. What the shell theorem does say is that for an object with a spherical mass distribution (density is a function of radial distance from the center), it's only the mass below that counts. You can use the shell theorem to find the condition that make gravitational force increase or decrease with increasing depth. Defining ##\rho(r)## as the density at some distance ##r## from the center and ##\bar{\rho}(r)## as the average density of all the stuff below that distance, you should find that gravitational force increases with depth if ##\rho(r) < \frac 2 3 \bar{\rho}(r)##.

Yeah I understand now what he was referring to just didn't connect the dots from his post lol.
 P: 3 wow, took me a too little long to wrap my head around all this for my liking :p thanks everyone for your input. guess i'm going to have to come up with a better idea then 'dig a big *** hole' to address the low grav issue. any input??
 Mentor P: 14,242 What low gravity issue? You are assuming that it is an issue, and you are assuming that humans will confront it. Those are both very big ifs. Regarding the former, while zero gravity is known to be problematic to humans, nobody knows where the cutoff is between 0g and 1g that delineates harmful from not harmful. Perhaps there's no big problem with 0.38 g. Perhaps it's even pleasant. Regarding the latter, that is an even bigger if. We may never send people to Mars. If we have the wherewithal to send humans to Mars long term, we would also be close to having the wherewithal to make a largish space station, one that is hospitable to people. Why would we want to go back down into a gravity well when we just spent an enormous amount of capital to get out of one? Another downside / risk is Mars life. It's a fairly good bet that Mars did support life of some form long ago. Whether it still does, nobody knows. If it does, there's a potential for harm both ways, Mars life harming us and us harming Mars life. There's a significant faction in NASA (some rather high up) and elsewhere that think that humans must *never* go to Mars if Mars does support life.
PF Gold
P: 1,368
 Quote by D H Regarding the latter, that is an even bigger if. We may never send people to Mars. If we have the wherewithal to send humans to Mars long term, .
There is a dutch company that is asking for volunteers, though its a one way trip. scheduled for 2023.