\(x^{ 2 }=r^{ 2 }_{ e }+r^{ 2 }_{ e }-2r^{ 2 }_{ e }cos(\phi )=2r^{ 2 }_{ e }(1-cos(\phi ))\)
\( x=r_{ e }\sqrt { 2(1-cos(\phi )) } =2r_{ e }sin(\phi /2)\)
and
\( \cfrac { r_{ { T^{ + } } } }{ x } =tan(\theta _{ 0 })\)
\(\cfrac{ r_{ { T^{ + } } }}{tan(\theta _{ 0 })}=2r_{ e }sin(\phi /2)\)
\(\phi=2.sin^{-1}\left(\cfrac{ r_{ { T^{ + } } }}{2r_{ e }tan(\theta _{ 0 })}\right)\)
This is the relationship between \(\theta_0\) and \(\phi\), as \(x\), the distance between the electron and the center of the \(T^{+}\) particle's orbit changes.
Low values of \(\theta_0\) corresponds to \(T^{+}\) particles that are far away, (along \(x\)), from the electron. When \(r_e\) is finite,
\(\theta_{0\,\,min}=\theta_c=tan^{-1}(\cfrac{r_{\small{T^{+}}}}{\sqrt{2}.r_e})\)
Each orbiting \(T^{+}\) particle generates a \(E\) field perpendicular to it orbital plane, at an angle \(\pi/2-\phi_s\) to the reference \(\phi=0\).
all these oscillations overlap at the lower values of \(\theta_0\), beside \(\theta_c\).
But the traveling \(T^{+}\) particles has high velocities at these values of \(\theta_0\) near \(\theta_c\), the center of the oscillation; the equilibrium position. The particles pass through \(\theta_c\) quickly but dwell at the two ends of their oscillations.
If there is an uniform spread in oscillations from \(\theta_c\) to \(\theta_0=\pi/2\), there will be a greater concentration of \(E\) fields from higher values of \(\theta_0\), due to the \(T^{+}\) particle longer dwell time. (A math expression is needed here.)
This could explain why higher values of \(\theta_0\) has higher energy and is associated with higher frequencies and so lower wavelength.
In this case however, \(T^{+}\) is oscillating along the electron orbit with frequency \(f_s\), and is also revolving in its orbit perpendicular to the electron orbit at frequency \(f_o\). The concentration of \(E\) field is due to both \(f_s\) and \(f_o\). The spread of energy (sum of all overlapping \(E\) fields) however is due to \(f_s\) only.
If the traveling \(T^{+}\) particles has lower velocities (ie. lower energy oscillations), and their dwell time at higher values of \(\theta_0\) is insignificant compared to the overlap, then values of lower \(\theta_0\) will have higher energy and be associated with lower wavelength. The spectrum is reversed for low energy oscillations. This could explain why the emission spectrum is reversed at lower temperature.
Previously, the half angle of a light cone which give the strongest \(E\) field when at \(\pi/2\), (c/f post "It's All Fluorescence Outside, Inside" dated 29 Jul 2015.), the angle also defined as \(\theta\) was polarization, not wavelength (color).
Good Night...
