Moving Trough

The electron orbital radius,  $r_e$  was obtained from equating forces, together with the drag of free space at near terminal velocity.   $r_e$  was then made to vary with temperature,  $T$ as the drag factor,  $A$ varies with  temperature.  For all values of  $r_e$,  the resultant force on the electron is zero.

$\because \sum F=0$

Force per unit inertia,  $F=-\cfrac{\partial\,PE_{re}}{\partial\,r_e}$

as in

$E=-\cfrac{d\,V}{d\,x}$

So,

$\cfrac{\partial  PE_{re}}{\partial r_e}=0$

 As such its potential,  $PE_{re}$  is at an extrema for all values of  $r_e$.    And we understand it to be a minimum as the electrons are in stable orbits.  (If it is a maxima, a small perturbation will result in a positive force that is the negative of the gradient, pushing the electron further away from the extreme point.  ie unstable.)  This potential trough moves along the  $r_e$  vs  $T$  profile as  $T$  changes.

In other words,  $PE_{re}$  is always a minimum on the  $r_e$  vs  $T$  profile.

Have a nice day.