The graphform,
\(en(\theta)=\cfrac{sin^3(\theta)}{cos^2(\theta)}e^{-\cfrac{4r_e}{a_ecos(\theta)}}\)
defines the envelope of the radiated energy.
\(\cfrac{d\,en(\theta)}{d\,\theta}=tan(x){ e }^{ -\cfrac { 4r_{ e }sec(x) }{ a_{ e } } }\left( sin(x)+2tan(x)sec(x)-\cfrac { 4r_{ e } }{ { a }_{ e } } tan^{ 3 }(x) \right) \)
A plot of this graph whose zeros are the maxima of the envelop, with \(\cfrac{r_e}{a_e}\) varying from 1 to 2, is shown below. There are three zeros, two points of inflection at 0 and π/2 and one maximum where the gradient of the derivative is negative.
For the maximum points,
\(sin(x)+2tan(x)sec(x)-\cfrac { 4r_{ e } }{ { a }_{ e } } tan^{ 3 }(x)=0 \)
For the points of inflections,
\(sin(x)=0\)
Do not steal my money. No discontinuity. We are dealing with radiated energy from varying magnetic field. Previously when we encountered a discontinuity in the gradient of \(r_e\) vs \(T\) we were dealing with electrostatic, centripetal force and thermal gravity, even gravity itself.
May be there is a similar bandgap, a magnetic bandgap that is not visible here.
