Sphere packing with inert bubbles

In natural units, where , the expression becomes .

This shows that the gravitational coupling constant can be thought of as the analogue of the fine-structure constant; while the fine-structure constant measures the electromagnetic repulsion between two electrons, the gravitational coupling constant measures the gravitational attraction.

Now that I see this, I want to revise my interpretation of the fine structure constant.  I am fishing here, so beware.  Gravity is  about how much motion bubbles of a single size need to measure 4*Pi exactly when filling the balloon.  Thus the fine structure is about how much motion is required when filling the balloon with two sizes of bubbles.  But something is wrong because the difference between the fine structure and the gravity structure is about 10e-45. So, there may be another Avogadro stuck somewhere between the two, I am not sure; an Avogadro squared is about 10e45. In terms of relative sphere size, the charge spheres, two of them, and the inert sphere cannot be that far apart. The Swarzchild radius gives the coupling constant when the balloon has all three bubble types:

 where p is the density.


The Sun is so round!

Yes, but it is emitting enormous excess energy, it has more motion than needed to get a good 4*pi.  This is still mostly a puzzle for me.

Inert spheres have no motion

Yes, but when optimally packed the variation in surface area should be enough to get the best 4*pi possible.  In fact, I think Gauss already did this problem.
 I am pretty sure physics is getting down to the business of sphere packing.

So all you sphere packing mathematicians, gear up, your skills will be in demand.