I just noticed yet another gaffe in The Core.

(SPOILERS AHEAD)

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Our intrepid heroes, in their nearly-indestructable Earth Ship, tunnel all the way down to the molten outer core of the planet. Yet, while they're walking around on the deck of their ship, they're still feeling the full 9.8 m/s^2 of Earth's surface gravity. (None of them, not even the wacky sidekick characters, exhibits any clumsiness while walking around. Two characters even get pinned underneath 6-foot-tall thermonuclear devices because said devices are "too heavy" for them to lift.)

If you were down that close to the center of the planet, most of the Earth's mass would be "above" you, i.e. farther away from the center of the planet that you were. None of that mass should contribute to the force of gravity you feel. Only the mass beneath you should "count." Why, then, weren't our intrepid heroes running around in 1/2 or even 1/3 Earth normal gravity?

Umm..The unobtanium produces its own localized gravity field?

"Journey to the Center of the Earth" had it more accurately than The Core. They at least realized that gravity would go all wierd near the center of the planet except they should have had to go up through a layer of water to be on the surface of it if I am thinking correctly. -Colt

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Actually, Journey to the Center of the Earth , while one of my favorite movies, had it less accurately than "The Core". In Journey to the Center of the Earth, besides the ridiculousness of an ocean in the core, the only effects that could be considered weird gravity were when they actually reached the center of the Earth. Here, where they should have been weightless, the gravity did not change, but somehow they knew that the center of the Earth was a place that attracted gold, and all the gold on the raft flew off. No explanation was given as to why the core attracts gold- I think this was an old idea like the ether or alchemy.

Actually they wont be weightles at the core. Most of the mass of the earth is above them. They would be walking on the ceiling.

But there is equal mass pulling at them from all directions, so they should be weightless (maybe not absolutely since the Earth isn't a perfect sphere).

Ahhh...I get it. But that would be only if they are in the exact center. What will happen if they are slightly off center? Would they not slowly (very slowly) start to fall towards that part of the planet?

If they were off center they'd fall in the direction of greatest mass (however slowly). But remeber there is an equal amount of mass on the other "side" of the core, so they'd fall to the center with decreasing weight (anyone correct me if im wrong) and then when they get to the center they would become wieghtless.

In the movie, they never went all the way to the center of the Earth. They stopped while they were in the outer (molten nickel-iron) core.

Most, but by no means all, of the Earth's mass would have been "above" their depth at that point. "Down" would still have been toward the center of the Earth, but their weight should have been a lot lower than it was at the surface.

For an example say a group of scientists are drilling to the core of a planet (this planet happens to be a perfect sphere and alll mass and densities are all uniform throughout the entire thing) 1000 km in diameter making its radius 500 (duh). If you were at the center all the mass in the 500 km to the surface in every direction will be pulling you each way so you would be weightless. If you drill from the surface 480 km into the planet with 20 km left to go to the center you will feel the affects of 40 km pulling you towards the center because the 480 km you have traveled (which would be "up) cancels out 480 km of the 520 km depth directly ahead of you from you to the opposite point on the surface from where you started out( "down"). So the 20 km of mass between you and the center would be added to 20 km of another radius (or the second half of the diameter) from the center to the opposite point on the planet's surface form which you started drilling that didn't get canceled out by the 480 km behind you. The point is even if you've traveled almost all of the radius you've traveled a a little less than half of the diameter meaning that you can never fall "up".

But the writer said the science was dead-on! I can't believe Hollywood would lie to me like that!

Pretty much, yeah. However, attractive force increases according to the inverse square law. So, since they're closer to the core of the Earth, they'd be experiencing more attraction in that direction. I'm too lazy right now to work out the numbers of falling-off radius versus falling-off mass, but since the mass distribution of the Earth isn't uniform, it could make for some whacky gravitational effects.

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I doubt that the interior of the Earth would be so nonuniform that it would make gravitational effects that were all that wacky -- until you started tunnelling into the inner (solid iron-nickel) core and got really close to the geometric center of the planet, which they didn't do in the movie.

The Earth started as a big molten ball of metal and rock and cooled over hundreds of millions of years before it had a comfy solid surface on it. That was plenty of time for all the dense stuff to settle down at the center and all the less-dense stuff to "float" to the top, and the distribution is probably pretty uniform. [I'm guessing it was this "densest stuff sinks to the center" tendency that inspired what's-his-face to hypothesize a giant nuclear fission reactor in the Earth's inner core -- solid Uranium under normal conditions of temperature and pressure has a specific gravity of 19.05, which puts its density only slightly behind tungsten (19.25) and gold (19.3), and a weensy bit farther behind rhenium (21.02), platinum (21.09), osmium (22.61), and iridium (22.65). If the densest stuff really does sink to the center as a uniform, compacted layer, then there should be a thin, uniform spherical shell of uranium near the center of the inner core surrounding a (perhaps thicker) spherical shell of tungsten, and so on in concentric shells of greater density as you go downward, until you hit iridium at the dead center.]

Nope. It's true, they are exactly that wacky.

As much as it pains me to say this, The Core got that science right. Of course, it could have been by accident, since the force of gravity at the top of the core is very nearly the same as the force of gravity on the surface of the Earth. In fact, it's pretty much constant all the way down to the top of the core--increases a little bit in fact--so they didn't have to do anything to get it right. (That seems to be in keeping with the rest of the movie. )

The core is more or less twice as dense as the entire Earth, and has a radius of about half the Earth. The force of gravity is proportional to the mass divided by radius squared, but mass is proportional to density times the cube of radius. As has been pointed out, the mass above your radius can more or less be ignored. So the force is proportional to density times radius, and since one is twice and the other half, the result is that the gravity at the core is about the same as at the surface.
The dense stuff does not sink to the bottom. The upper layer of the Earth has a much higher concentration of radioactive elements than the lower layers.

The definitive word:

You can basically consider the Earth to be of uniform density for our purposes. As you headed toward the center of the Earth, the acceleration due to gravity would decrease linearly according to the equation:

a = G*4/3*pi*r*rho

Where rho is the density and r is the radius.

If they were halfway to the core then they should experience half the gravity. At the core (r=0) there would be no acceleration due to gravity at all.

Three points. If you get 12 points you lose your license.

Sorry, but I gotta agree with kilopi on this one.

Your formula is correct, JS Princeton, but only if rho (the density) is constant all the way down. It clearly is not -- the Earth's average density is some 5.5 grams per cubic centimeter, but according to this webpage, the average density of the entire core is 10-12 grams per cubic centimeter, or about twice the average density of the Earth as a whole.

According to this U.S. Geological Survey page, the top of the outer core is about 2900 km below the surface of the Earth. Since the Earth is 6378 km in radius, this makes the core's radius equal to 0.55 times the radius of the Earth, which would make the core's volume equal to 0.17 times the volume of the Earth. (The volume of a sphere is directly proportional to its radius cubed.)

Now then, the general formula for gravitational acceleration in Earth g's is M/r^2, where M is the mass beneath you (in Earth masses) and r is your distance from the center (in Earth radii). Assuming a volume of 0.17 Earth volumes and a density of 2 Earth densities , the core would have a mass M of 0.34 Earth masses.


Therefore, when our intrepid heroes in the movie were at the tippy-top edge of the outer core, they should have been experiencing gravity of 0.34 / 0.55^2 = ... holy cats ... 1.12 times normal Earth surface gravity!



I take back everything I said at the beginning of this thread. They would weight slightly MORE at the surface of the core, not less!!

Mea culpa, mea culpa, mea maxima culpa!

So therein lies the point. If you multiply rho by 2 and divide r by less than 2 you end up with more gravity.

Us fly-by-night approximators need to be clear about our assumptions. My assumption that the average density was good enough was wrong, dead wrong.

However, what I don't get is that the Cavendish experiment measures a mean density of 5 g/cc. Where on "Earth" does this number come from? After all, with a radius of 6300 km and a mass of 6.E27 grams, you get a much higher mean density.

Thanks for the 3 points, kilopi, I'll store them in my pocket.

Nine Planets gives those figures as 6378.2 km and 5.972E27 grams. That's close to what you got, and using them I get 5.495 gr/cm^3, which is very close to the 5.5 that I've always used--and is the one tracer used.

And I just remembered, we've talked about this before.

Let us first assume that the horse is a massless, frictionless sphere....

(From some joke about theoretical physicists going into horse racing.)

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Right, so the number is right, for the average, does that mean that the mantle in the crust have a lower density than 5 g/cc?

Weighting the volume with 17%, and considering the core to have a mean density twice that of the average, that means the outer density of the mantle and the crust is closer to 4 g/cc.

Ah the places you'll go.

Yep, that sounds about right.

Of course, in the movie, when they finally reched the outer core, they discovered that its density was less than we had previously estimated. If the cores average density was only 8 g/cm^3, our intrepid heroes should have been feeling a gravitational force of only about 0.8 times Earth normal surface gravity.

Density of granite is less than 3g/cc

And because there mantle contains more iron, magnesium, and calcium than the crust, and is also denser due to being compressed (pressure inside the Earth increases with depth), a mantle density above that of granite is to be expected.

The crust is mostly silicate, SO2 or SO4 etc.

I hope you mean SiO2, not SO2. I'd hate to have to worry about breathing sulfur dioxide every time I walked around a patch of silicate!

So, given that gravity decreases to (very close to) zero as we approach the center of the Earth, wouldn't that mean that pressure also decreases? Wouldn't heat be reduced as a result? If we could get through the initial 3900 miles, wouldn't the last leg be a cake walk?

I'll bet the center of the Earth (or that of Jupiter for that matter!) could be a very cool place!

Heh. No. The lack of "surface gravity" if you "stand" at a particular point within an object doesn't do a whit to reduce the pressure from all the material above you, which will all be experiencing higher gravity.

The center of the sun is at zero-gee, but it's hot enough and high-pressure enough to fuse protons into alpha particles.

LOL. Yep, I guess I was too focussed on not being able to subscript.

You'd think on a message board devoted to astronomy, you'd have formatting tags to do proper exponents and chemical formulas. Heck, vBulletin and UBB both support [sub] and [super] tags.

Okay, I just saw The Core for a second time. (I must have a masochistic streak in me.)

This time, I noticed something I missed before. When they were bench-pressing the nuclear (excuse me, nuke-you-lar) warheads to move them into different compartments, the Virgil was a lot farther down than I previously thought. They were skimming along the top of the inner core.

That changes the gravity equation. The radius of the inner core is only 0.2 times the radius of the Earth, giving it a volume equal to a mere 0.0080 times the Earth's volume. Assuming that the inner core is twice as dense as the Earth as a whole, this would make its mass equal to 0.016 Earth masses.

Therefore, when our intrepid heroes in the movie were at the tippy-top edge of the inner core, they should have been experiencing gravity of 0.016 / 0.2^2 = 0.4 times the surface gravity of the Earth!

This is about the same gravity as you'd experience standing on the surface of Mars. So, like I asked in the first post in this thread, why weren't they walking funny or having an easier time picking up those nuclear bombs?