[sgdpk]

As someone commented, this example is not so mysterious. It's just a change of variables to make the argument of exp() dimensionless.

Here's a good example of the power of dimensional analysis: how small should the Earth be to collapse into a black hole?

Knowing nothing about the problem, you know it should involve at least two things: the mass of the Earth, M, and the gravitational constant G.

Since F=ma and F=GMm/d^2, we know that GM has units of distance^2 * acceleration (check that). This is equal to distance * (distance/time)^2.

We want a radius, which is a distance. And we almost have it! At least if we can get rid of the (distance/time)^2 factor. But that's a velocity^2! Now, what's a velocity that should be natural in questions about black holes and general relativity? Why the speed of light c, of course.

So, we can guess that the answer is GM/c^2.

Now compare this with the real answer: https://en.m.wikipedia.org/wiki/Schwarzschild_radius

[evanb]

In fact this can also be understood "just" as a simple change of variables in either Newton's law (as you have shown) or Einsteins' field equations, to dimensionless quantities!