Is there a working law of gravity in 2D?

I am constructing a two dimensional world, but ran into a problem with one of the fundamental forces, gravity.

I first tried to see what would happen if I just used the normal law of gravity, $\frac{m_1m_2}{r^2}$. As it turns out, it has a problem:

Consider a person standing on an edge of a "planet" in 2D. then, the area coloured green has an influence on you (I have made things with area have mass, in order for the concept to work in my 2D world.):

Now, consider another area with the same shape inside that one:

It has just $\frac{1}{2}$ the distance, thereby four times the gravity. But it also has just $\frac{1}{4}$ of the area, and therefore a total gravitational influence similar to the green area. I can continue to stack those areas, with a sum of 1+1+1+1... Every edge then becomes a black hole!

The same argument does not hold for 3D, as half-shells have an influence of $1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}...$ which does not reach infinity. (You can obtain that result yourself by noticing that your skin is not made up of black holes).

Obviously, I must then choose another exponent for $r$, and that can not be 2 or larger, because of the half-circle argument, and it can not be 0 or lower, as that would make the whole universe collapse into a black hole.

Also, I want orbits to be periodic, to make planet systems stable. I know $exp=1$ does not have periodic orbits other than in special cases, but there must be other possibilities than $exp=2$, as the case $exp=-1$ is periodic.

Is there a solution with periodic orbits for $0

posted over 8 years ago

SE - stop firing the good guys‭

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