\(ln(cosh(\cfrac { G }{ \sqrt { 2{ mc^{ 2 } } } }a_{\psi\,c}))=\cfrac{1}{4}\)
\(ln(cosh(0.7369))=\cfrac{1}{4}\)
and
\(ln(cosh(\cfrac { G }{ \sqrt { 2{ mc^{ 2 } } } }a_{\psi\,\pi}))=ln(cosh(\pi))=2.450311\)
what is between \(a_{\psi\,c}\) and \(a_{\psi\,\pi}\), where the force increases with distance away from the center?
An accumulation of other like particles. At distances greater than \(a_{\psi\,c}\), up to \(a_{\psi\,\pi}\), the attached particle experiences a greater attractive force due to the increasing value of \(F_{\rho}\).
\(a_{\psi\,c}\) is then interpreted as the minimum value for \(a_{\psi}\) that holds up a particle. Values of \(a_{\psi}\) less than \(a_{\psi\,c}\) collapses the particle.
And we have a lumpy issue...Particles of various sizes that stick together. Interestingly,
\(F=\int{F_{\rho}}dx\)
at \(x=a_{\psi\,c}\),
\(F=m\cfrac{c^2}{2}\)
This is the attractive force that holds the particles in the lump together.
