A hypothetical question about gravity

Subductionzon

Of course to do this you can only do this if you ignore certain physical laws. If it was a hole through the actual Earth your acceleration would increase for the first half of the trip until you hit the core/mantle boundary at which point you would be accelerating at about 10.8 m/sec^2 if there is a vacuum in the hole.

Now if there is air in the tube the pressure will go up until the air is no longer a gas, it would be a supercritical fluid. If the hole was man sized it will not hold a significant fraction of the Earth's gravity.

So let's put magic baffles in the tube. We already have magic sides that keep the heat out and stand up to amazing pressures so a little more magic can't hurt. You will fall at terminal velocity the whole trip down and never quite reach the center.

Ken G

Gold Member
I meant to say it has an upwards component too.
But it doesn't, all the force from mass outside the radius of the object will add to zero. You only need to look at the mass of the sphere inside that radius, and ignore the rest.
Yes the density will increase but I doubt it would be much at all because there probably isnt enough air in the atmosphere to fill the hole if its wide enough.
The hole doesn't need to be kilometers wide, so there's no need to worry about that.

Ken G

Gold Member
Of course to do this you can only do this if you ignore certain physical laws. If it was a hole through the actual Earth your acceleration would increase for the first half of the trip until you hit the core/mantle boundary at which point you would be accelerating at about 10.8 m/sec^2 if there is a vacuum in the hole.
So you are including actual density variations within the Earth, we'll probably have to take your word for that calculation, given your handle!
So let's put magic baffles in the tube. We already have magic sides that keep the heat out and stand up to amazing pressures so a little more magic can't hurt. You will fall at terminal velocity the whole trip down and never quite reach the center.
Yes, the hole would have to be magically held open, or pressure would close it immediately. I agree you would probably never formally reach the center, but in practical terms, you probably would because the approach is exponential in time. It might take quite a while though, you'd probably starve to death waiting!

A.T.

Now if there is air in the tube the pressure will go up until the air is no longer a gas, it would be a supercritical fluid.
Would the pressure really be enough to make air liquid at the center? I guess it depends on the temperature. But let's assume the tunnel is well insulated.

Anyway, gas or liquid, it would be denser.

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Aero51

The acceleration of gravity will decrease (approaching zero) as you move towards the core. The net contribution will always face the center of the earth but there will be a component due to the mass behind the object that will account for the decrease in g.

Ken G

Gold Member
The acceleration of gravity will decrease (approaching zero) as you move towards the core. The net contribution will always face the center of the earth but there will be a component due to the mass behind the object that will account for the decrease in g.
But that's not a very useful way to think about what is going on there. The force decreases simply because there is less mass in the sphere within the radius of the falling body. There is no net contribution from the mass outside that sphere, so the force should not be thought of as falling off due to any contribution coming from out there.

Ken G

Gold Member
Would the pressure really be enough to make air liquid at the center?
No, not if we are magically holding the hole open. Then the pressure of the air would just come from the weight of the air in the shaft, not from the weight of the Earth, and it would not be any kind of huge pressure.

poco9964

Would the pressure really be enough to make air liquid at the center? I guess it depends on the temperature. But let's assume the tunnel is well insulated.

Anyway, gas or liquid, it would be denser.
I see no reason to assume there would even be these kind of pressure present in the hole. Mainly because all the mass that would have caused that pressure would have been removed when making the hole. The only pressure would be air, it would cause a displacement that could have catastrophic effects on the crust. It may effect the layers of the atmospheres that protects us from harmful UV rays. Ignoring that, I am not even sure if the air could be drawn to the centre. Mainly because its mass its significantly less than the object that is falling through it. The point at which air could resist the gravity and reach a state of equilibrium would be a lot higher up the hole than we could fall. I actually think it would create a void in the centre, that closely resembles space. And if we managed to reach it, it would be like being in a membrane that you have no friction to stop you bouncing of the sides, and not enough inertia to ever break free again. Well that is if you do not go down with something that packs the punch of a soyuz rocket. But fits in your pocket.

poco9964

But that's not a very useful way to think about what is going on there. The force decreases simply because there is less mass in the sphere within the radius of the falling body. There is no net contribution from the mass outside that sphere, so the force should not be thought of as falling off due to any contribution coming from out there.
Now that has got my mind thinking.

If I reached a point where the mass in the sphere above me was greater than that what was below me. Why does this have no effect would on my mass, and why would the mass above me now not become the greater force?

Sorry if that is a silly question.

A.T.

The only pressure would be air,
But a lot of it. 1 bar is from the 100km column above surface at approx. constant g. In the tunnel center the column would be 63 times higher, however at decreasing g. My question was if that is enough pressure to make air liquid.

It may effect the layers of the atmospheres that protects us from harmful UV rays.
If you don't make the tunnel to wide, I doubt it. The polar bears could reach the Antarctic and eat all the penguins.

Ignoring that, I am not even sure if the air could be drawn to the centre. Mainly because its mass its significantly less than the object that is falling through it. The point at which air could resist the gravity and reach a state of equilibrium would be a lot higher up the hole than we could fall. I actually think it would create a void in the centre, that closely resembles space.
What do you mean? What should stop the air from falling to the center?

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A.T.

Now that has got my mind thinking.

If I reached a point where the mass in the sphere above me was greater than that what was below me. Why does this have no effect would on my mass, and why would the mass above me now not become the greater force?

Sorry if that is a silly question.
The mass "above you" in terms of a spherical shell has no effect on you. The gravity of it cancels itself.

The mass "above you" in terms of a sphere section would only become greater if you pass the center. Then gravity is reversed.

mfb

Mentor
Would the pressure really be enough to make air liquid at the center? I guess it depends on the temperature. But let's assume the tunnel is well insulated.

Anyway, gas or liquid, it would be denser.
Close to the surface, air density doubles every ~8km. This would give conditions similar to the critical point (~3,5MPa, 126K for nitrogen) already 40km below the surface. Give or take a factor of 2, the air would be supercritical in most of the tunnel unless you evacuate it in some way.

D H

Staff Emeritus
A.T. and Subductionzon have it right. Gravitational acceleration inside the earth reaches a global max at the core mantle boundary and then drops to zero per the Preliminary Reference Earth Model.

Reference: Dziewonski & Anderson, Preliminary reference Earth model, Physics of the Earth and Planetary Interiors, 25:4 (1981)
DOI: 10.1016/0031-9201(81)90046-7
Tabular presentation: http://geophysics.ou.edu/solid_earth/prem.html [Broken]

A simple model of gravitational acceleration inside the Earth is that gravitation remains constant at 10 m/s2 from the surface to halfway to the center and then drops linearly from that point inward. This is much more amenable to analysis than is the PREM but still retains the key feature that gravitational acceleration remains high in the crust and mantle.

What happens to air inside the magical tunnel depends on temperature. Assuming this simple model, ideal gas conditions, hydrostatic equilibrium, and a constant temperature of 20 C throughout (magical walls!) would mean an absolutely ridiculously high pressure at the center of 5.5*10493 atmospheres. Assuming hydrostatic equilibrium and adiabatic conditions would mean ridiculously high pressure and temperature at the center.

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A.T.

would mean an absolutely ridiculously high pressure at the center of 5.5*10493 atmospheres.
I guess this ridiculous result is due to the assumption that the gas can be compressed without limits, so that density increases steadily?

The molecular hydrogen atmosphere of Jupiter is thicker than one Earth radius. Gravity is in the same order of magnitude in that region. The pressure below that atmosphere is estimated to be 200 GPa. Which is much, but nowhere close to 10493 atmospheres.

http://en.wikipedia.org/wiki/Jupiter

Subductionzon

Would the pressure really be enough to make air liquid at the center? I guess it depends on the temperature. But let's assume the tunnel is well insulated.

Anyway, gas or liquid, it would be denser.
Not liquid, that is impossible. Air is far above the critical temperature of the various gases that are in it. It would be as I said a supercritical fluid. I did the math once before when trying to calculate the density of the air using the Ideal Gas Law. I got a ridiculous solution that showed me the law broke down at those pressures.

Using the ideal gas law you don't have a gas column of constant density. You will have one that will continually increase in density. Eventually you are not dealing with a gas anymore. Since the force of gravity goes up slightly for the first half of your journey you can make the problem easier by treating it as a constant for that distance. It should double in density over a constant distance. At sea level the pressure drops off to half at 18,000 feet or 5,500 meters. Going down half way you would have 280 of these intervals. Or the density of the air, if the Ideal Gas Law held up would be 2^280 times that at the surface. This amount would obviously be in error so there is no use in using the Ideal Gas Law. Therefore the need of our magical baffles to keep the air at a reasonable pressure.

Subductionzon

I see D.H. did a more exact analysis that showed that the air has to be a supercritical fluid and no longer a gas. The reason that the pressure gets so high in the calculations is that the Ideal Gas Law assumes that air is infinitely compressible. As we go down the hole not only would the air be increasing in pressure due to the weight of all of the gas above us, the density of that gas would be increasing with that increasing pressure resulting in exponential growth to the pressure and density. Sooner or later the IGL will fail.

D H

Staff Emeritus
I did something dyslexic with my calculation, using R=3678 km rather than 6378 km.

In any case, an adiabatic assumption is arguably better than the isothermal assumption. This yields a temperature of about 48000 kelvin at the center of the Earth and a pressure of about 56 million atmospheres.

Details:
Gravity is constant at 10 m/s^2 down to the halfway point. This yields a lapse rate of 9.953 K/km (c.f. the adiabatic lapse rate of 9.8 K/km for g=9.80665 m/s^2). This makes the temperature at the halfway point be about 32000 kelvin. Below the halfway point gravity drops linearly toward zero at the center. This makes temperature quadratic below the halfway point,
$$T(z)=T(R/2) + \alpha*(z-R/2)\left(1-\frac{z-R/2} R\right)$$
where $\alpha$ is 9.953 K/km. This yields a temperature of 48000 kelvin at the center of the Earth. The assumption of adiabatic conditions means
$$P=P_0 \left(\frac{T}{T_0}\right) ^{\frac{\gamma}{\gamma - 1}}$$
Using $\gamma=1.4$ yields a pressure of 56 million atmospheres at the center of the Earth.

Bottom line: This air-filled tunnel is very hot and under extremely high pressure.

poco9964

But a lot of it. 1 bar is from the 100km column above surface at approx. constant g. In the tunnel center the column would be 63 times higher, however at decreasing g. My question was if that is enough pressure to make air liquid.

If you don't make the tunnel to wide, I doubt it. The polar bears could reach the Antarctic and eat all the penguins.

What do you mean? What should stop the air from falling to the center?
The same thing that causes a lack of Air in caves or mine shafts. I understand this hole will be opened both ends, but air will not flow from one end to the other. It would be like getting a pipe and attaching fans either end blowing inwards. They would reach a point where they would cancel each other out. In the centre, I would assume as it is equal forces being excreted. A better analogy would be to block both ends of a pipe so no air can escape, and then use a plunger mechanism on both ends to push them together. You will notice it takes tremendous force to do so. I can not see why the same effect should not occur in this hole.

The lack of gravity will also make air lighter than at the surface, so even though it got denser it does not mean it would get heavier. Gravity would begin to have less of an effect on its mass, the same as it did our own. And at zero point it will have no effect on its mass.
It's increase in density will increase air resistance the further you go down the hole. Yet another factor that I feel would prevent you reaching the core. Also the hotter it got, the more updraught would be created like thermals. giving even more reason as to why the air, like us would not make it to the core. Even if you could control the heat at the core, you would struggle to control the heat generated under the air pressure.

mfb

Mentor
The same thing that causes a lack of Air in caves or mine shafts.
Can you explain this, please?
I understand a lack of oxygen (as it is consumed by humans, and finite without ventilation). But I never heard of underpressure without sealing and pumping.

It would be like getting a pipe and attaching fans either end blowing inwards.
This pipe will have a higher pressure inside.

A better analogy would be to block both ends of a pipe so no air can escape, and then use a plunger mechanism on both ends to push them together. You will notice it takes tremendous force to do so.
This is a good sign of a high pressure.

The lack of gravity will also make air lighter than at the surface, so even though it got denser it does not mean it would get heavier.
Weight gets lower, density increases. This means that a lot of air is in the center.

poco9964

Can you explain this, please?
Sorry, I have not explained myself very well. I was trying to explain the convection effect, that heat makes molecules move faster. And that the force of gravity is temporarily over come all the time these molecules are excited enough. When they cool down gravity grabs them again and brings them back down. This effect will actually prevent the air density becoming any denser. The air will reach a certain point where the heat generated from the pressure above it prevents it going any lower due to the convectional heat (Pressure) being generated beneath it. As gravity gets weaker and the air gets hotter, there becomes less chance they could ever meet. I can not do the maths, but I would expect that to occur long before gravity started to drop off. I hope I have made myself a little clearer.

"A hypothetical question about gravity"

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