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Thank you.

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Near the ground the atmospheric density drops to 1/2 every 5 kilometers (8 kilometers = 5 miles, very nearly), so at 10 km where jets fly, density is down to 1/4. This continues more or less up to 100 km (the halving distance varies a bit, with temperature), where collisions between molecules become relatively rare. Higher up the oxygen and nitrogen each decrease at its own rate.

The space shuttle flies at 300-400 kilometers, but even there enough air remains to seriously limit orbital lifetime. Also, enough of that air is ionized--electrons ripped off molecules by the Sun's extreme ultra violet, leaving behind positive "ions"--to reflect radio signals. Satellites orbit at 600-1000 km and up, and that, too, is where the first signs of the radiation belt can be observed, particularly off the Atlantic coast of Brazil, where the magnetic field is relatively weak.

Somewhere between here and there, you enter "space. "

(1) My students had a couple of questions that I thought were interesting. I told them I'd ask ya'll.

1. What would you weigh is you were at the exact center of the earth?
   
2. What would you weigh is you were 3 meters from the center of the earth?

Please include supporting evidence for your answers

(2)

I am confused about Newton's discussion of the force on a particle within a sphere in the Principia. In one place, he says that the force would be zero, since the attraction of all the particles in the sphere would cancel each other out.

Just a little further on, he says that the force would be directly proportional to the particle's distance from the center of the sphere.

Can you clarify these two seemingly conflicting statements?

Thank you for any light that you can shed.

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    Yours is an old question, first tackled by Newton, as the "Hollow Earth Paradox." If the Earth where a hollow sphere (inner and outer surfaces spherical) and someone dug a hole that reached the hollow interior, and then stepped into it--what would that person experience?

    Newton's answer--there would be no gravity inside the hollow. Any object thrown into it--say, a stone--would continue in a straight line with constant velocity (ignoring air resistance).

    Newton's argument was roughly as follows. Take an object at a point in space anywhere in the cavity and draw from it a double cone (like a teepee, extending to both sides). Each side of it will cut part of the sphere, and the gravity of the two parts will tend to pull the object in opposite directions (make a drawing and you will see).

    Newton showed that the pulls of both part cancel each other: one part may be closer, but then it will also be smaller. Since all directions can be covered by a collection of such cones, the total force is zero. Today we get this result much more quickly by the theory of the potential, but that takes three-dimensional calculus, which Newton did not have.

    Now: Imagine you are somehow in the middle of the solid Earth--by some magic, not crushed by the rocks, suffocated or incinerated. In your mind you can divide all matter on earth into two parts: a smaller sphere containing everything that is CLOSER than you to the Earth's center, and a hollow sphere containing everything that is MORE DISTANT.

    By Newton, the hollow sphere exerts no pull, while the interior sphere, like the Earth, pulls as if all its mass were concentrated in the middle (that's another thing easily shown from potential theory). If you are halfway to the center, and the density everywhere is the same (actually, matter gets compressed towards the middle) then only 1/8 of the Earth mass is pulling you, but at half the distance, the pull is 4 times stronger ("inverse squares law"), so the final result is 1/2 of the gravity on the surface. At 1/N times the radius, the pull of gravity is just 1/N the pull at the surface.

    As you get deeper and deeper, the inside sphere gets smaller and its pull is weaker, so gravity too weakens. At the center, it is zero. At 3 meters from the center, it is the pull of a 3-meter sphere of rock, experienced on its surface--the pull of a tiny asteroid.

    Please note--that is just the pull of gravity on YOU. The rocks above you are also all pulled down, all the way to the surface of the Earth, and their weight is likely to crush you before you get very far. There may perhaps exist a cave a mile deep, but if so, none is much deeper, because there is too much weight piled on top.

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    The cheap and simple way is to build a shelter--especially since the dangerous events are the ones of solar outbursts, which are rare and last a day at most. You can calculate the shielding, but 20-50 cm rock should do a pretty good job (remember gravity is weaker, too, they will weigh less than on Earth).

    Have you read Ben Bova's "Mars"? It's fanciful science fiction, but his physics seems OK. The Mars astronauts are hit by a solar outburst halfway to Mars and wait it out, huddled in a special shielded area of their spaceship.