Newton demonstrated that the gravitational force at a distance r from the center of a uniform sphere is "as if" all the mass was concentrated at the center. The proof I have seen starts by first integrating over the 2d surface of an infinitely thin shell, and thus finding the gravitational field of such a shell. Outside the shell the above "as if" is demonstrated, and then since it is true for all concentric shells, it is true for the sphere.
But Newton's proof for a shell also proves that the gravitational field INSIDE the shell is exactly zero everywhere internal to the shell. That is, when the gravitational field of all parts of the shell are added up, you get identically zero.
Thus if you are in a deep mine shaft the gravitational field is only dependent on your distance from the center of the earth. That is, that portion of the mass of the earth in the shells above you has no effect on the gravitational field you experience. Further analysis says that if it were possible to drill a hole to the center of the earth, the gravitational field will be directly proportional to the linear (not inverse square) distance from the center of the earth. In particular, the gravitational field at the center of the earth would be zero, since all the shells above it have no net gravitational effect.
If the above argument is valid, why it is commonly said that the center of a sun is very highly compressed by the gravity of the sun's mass, and therefore hot enough to cause nuclear fusion? I have never understood this conclusion. Can someone explain it to me?