To answer these last three posts at once, hopefully, the integration for a disk plane does indeed come out to a result of infinite acceleration at the very edge for a point particle. A larger sphere at the edge, half in and half out should experience the same thing, at least for the point at its center, or considering the sphere a point mass at the center. In other words, if I were to perform separate integrations for every point in the sphere toward the disk, the point in the center would still blow up regardless of the rest of the points, and other points in the sphere shouldn't make much difference adding or subtracting with their finite contributions, then, I wouldn't think.

If we took a bunch of spheres and arranged them in a disk, then they would have thickness, and the acceleration toward the disk would be finite.

The computer program doesn't blow up at the edge, since it is calculating over clumped points within the disk instead of a mass spread evenly throughout the disk, as with an integration. However, as I make the thickness of the disk smaller and smaller, the gravity at the edge climbs higher and higher.

I know it doesn't seem to make much sense that the gravity at the edge of a finite mass disk should be infinite, but I have two theories about this. The first is with the thickness. With zero thickness, the particle does not lie against a surface as it would with a sphere or thick disk. Since the force varies with the square of the distance, we might need two dimensions to cancel out the infinities. That can be tested by finding for the potential energy, which only varies with the inverse of the distance, should become finite against the one dimension of the disk's edge. The other is with the clumping. I'm thinking stars, even gas and dust, are not spread evenly throughout a geometry, so the minimum possible distance between points must be taken into account.

I am wondering about the precise relationship between the resulting force and energy in all of this.

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"Let's define another operator, Sz, which we won't pay any attention to."
"This transformation will automatically make zero equal zero."
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